computational investigation of inverse heusler compounds

PHYSICAL REVIEW B 98, 094410 (2018)

Computational investigation of inverse Heusler compounds for spintronics applications

Jianhua Ma,1,* Jiangang He,2 Dipanjan Mazumdar,3 Kamaram Munira,4 Sahar Keshavarz,4,5 Tim Lovorn,4,5 C. Wolverton,2

Avik W. Ghosh,1 and William H. Butler4,5,†1Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, Virginia 22904, USA

2Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA3Department of Physics, Southern Illinois University, Carbondale, Illinois 62901, USA

4Center for Materials for Information Technology, University of Alabama, Tuscaloosa, Alabama 35401, USA5Department of Physics and Astronomy, University of Alabama, Tuscaloosa, Alabama 35401, USA

(Received 23 December 2017; revised manuscript received 28 May 2018; published 10 September 2018)

First-principles calculations of the electronic structure, magnetism, and structural stability of inverse Heuslercompounds with the chemical formula X2YZ are presented and discussed with a goal of identifying compoundsof interest for spintronics. Compounds for which the number of electrons per atom for Y exceed that for X andfor which X and Y are each one of the 3d elements, Sc-Zn, and Z is one of the group IIIA-VA elements: Al, Ga,In, Si, Ge, Sn, P, As, or Sb were considered. The formation energy per atom of each compound was calculated.By comparing our calculated formation energies to those calculated for phases in the inorganic crystal structuredatabase of observed phases, we estimate that inverse Heuslers with formation energies within 0.052 eV/atomof the calculated convex hull are reasonably likely to be synthesizable in equilibrium. The observed trends in theformation energy and relative structural stability as the X, Y , and Z elements vary are described. In addition tothe Slater-Pauling gap after 12 states per formula unit in one of the spin channels, inverse Heusler phases oftenhave gaps after 9 states or 14 states. We describe the origin and occurrence of these gaps. We identify 14 inverseHeusler semiconductors, 51 half-metals, and 50 near-half-metals with negative formation energy. In addition, ourcalculations predict 4 half-metals and 6 near-half-metals to lie close to the respective convex hull of stable phases,and thus may be experimentally realized under suitable synthesis conditions, resulting in potential candidates forfuture spintronics applications.

DOI: 10.1103/PhysRevB.98.094410

I. INTRODUCTION

During the past two decades, the field of magnetoelectronicsand spintronics has experienced significant development [1–3].Half-metals have the potential to play an important role in thecontinuing development of spintronics because they have a gapin the density of states (DOS) at the Fermi energy for one ofthe spin channels, but not for the other, creating an opportunityto achieve 100% spin-polarized currents [4,5]. A half Heuslercompound inspired the term “half-metal” when, in 1983, deGroot and collaborators calculated the electronic structure ofNiMnSb and recognized its unusual electronic structure [6].Since then, many Heusler compounds have been analyzed andhave been theoretically predicted to be half-metals or near-half-metals [7–12].

Due to the near 100% spin polarization at the Fermi level andthe relatively high Curie temperatures shown by some Heuslercompounds [13–15], half-metallic Heusler compounds havebeen exploited for devices [16], such as electrodes for magnetictunnel junctions (MTJs) [17–22], electrodes for giant magne-toresistive spin valves (GMRs) [23–26] and for injection ofspins into semiconductors [27].

Heusler-like compounds, sometimes called “inverseHeuslers” (IH) which are described below, can have electronic

*[email protected]†[email protected]

structures similar to those of the full Heusler and half Heuslercompounds, in the sense that they may be half-metallic orsemiconducting. In addition, the IH compounds can have anelectronic structure that has been described as a “spin-gaplesssemiconductor (SGS)” [28], i.e., a large gap at the Fermi energyfor one spin channel and a very small gap at the Fermi energyfor the other channel.

In this paper we will discuss our survey of IH compounds.We will concentrate on their bulk properties. For many appli-cations, heterostructures with interfaces will be needed. Theseinterfaces will affect the electronic structure and transportproperties, but are outside the scope of this paper.

The family of Heusler compounds has three subfamilies:the full Heuslers, the half Heuslers, and the inverse Heuslers.All three of these subfamilies are based on decorations ofatoms and vacancies on an underlying bcc lattice and are mosteasily visualized by beginning with a bcc lattice representedby the familiar cubic cell shown in Fig. 1(a) with all X-typeatoms. If the center atom is made different from the atomsat the corners, the structure becomes the B2 structure withcomposition XY [Fig. 1(b)]. If, in addition, alternate atoms atthe corners in Fig. 1(b) are made different, as shown in thelarger cell of Fig. 1(c), the structure becomes the full Heusleror L21 structure with composition X2YZ. The half Heuslerstructure (composition XYZ) can be visualized by removinghalf of the X atoms from Fig. 1(c) as shown in Fig. 1(d). Theinverse Heusler or XA structure (composition X2YZ) can be

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JIANHUA MA et al. PHYSICAL REVIEW B 98, 094410 (2018)

FIG. 1. Schematic representation of full Heusler L21 structure and inverse Heusler XA structure. (a) The body-centered cubic (bcc) structure.(b) The B2 cubic structure. (c) The L21 structure consisting of four interpenetrating fcc sublattices. (d) The half Heusler C1b structure consistsof three interpenetrating fcc sublattices. (e) The XA structure consists of four interpenetrating fcc sublattices. In the XA structure, X1 and X2

are the same transition-metal element but they have different environments and magnetic moments.

obtained from L21 by switching one of the X atoms in theL21 structure with a Y or Z as shown in Fig. 1(e). In all threefamilies, the X and Y atoms are typically transition metals (atleast for the systems that are potential half-metals), while theZ atom is typically an s-p metal.

We have recently completed a survey of the calculatedproperties of the three subfamilies of Heusler compounds. Theresults of these calculations are available in a database [29].We have previously reported on our study of the half Heuslercompounds [30]. In this paper, we describe the results ofcalculations of the properties of IH compounds with particularemphasis on those IH compounds that may be of use forspintronics applications and which may have a reasonableprobability of being synthesized.

A universal feature of the electronic structure of Heusleralloys that we consider is the existence of a relatively broad s

band followed at higher energy by bands that are primarilyderived from d states (Fig. 2). One particular feature ofthe electronic structure of the IH compounds is especiallyinteresting. The half Heusler and full Heusler families oftenhave band gaps after three states per atom counting from andincluding the just mentioned s band. We call this type of gap a

-10

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0

2(b) Spin-down

-10

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-4

-2

0

2

E -

Ef (

eV)

(a) Spin-up

7-9

10-12

13,14

11

2-62,3 4-6

7-910,11

12-14

K KL L

FIG. 2. The band structures of Mn2CoAl in (a) spin-up and (b)spin-down channels are shown. Bands from 1 to 14 are labeled in (a)and (b). The zero of the energy axis corresponds to the Fermi level.

Slater-Pauling gap because Slater and Pauling, but especiallySlater, noticed that many bcc-based alloys have approximatelythree states per atom in the minority-spin channel [31–33].Furthermore, Slater and Koster showed that for a nearest-neighbor tight-binding model for a B2 alloy (see Fig. 1) withonly d states, a gap forms in the center of the d band [31].

The “Slater-Pauling” state for bcc alloys has been ascribedto a minimum in the electronic DOS near the center of thes-d band. This minimum becomes an actual gap after threestates per atom in one of the spin channels in some Heuslercompounds. In contrast, although many IH compounds showSlater-Pauling gaps at three states per atom, some have gapsafter a number of states per atom that differs from three.

Thus, IH compounds often have gaps after 9, 12, and 14states per formula unit in the gapped channel. The gap after12 states corresponds to the Slater-Pauling gap. The gapsafter 9 states and 14 states are additional gaps allowed by thereduced symmetry of the inverse Heusler compared to the fullHeusler. Semiconducting states are possible with half Heuslersand full Heuslers, but these typically occur when both spinchannels have Slater-Pauling gaps, and both spin channels areidentical, implying no magnetic moments on any of the atoms.In contrast, the multiple gaps of the IH compounds allowsthe possibility of a magnetic semiconductor, e.g., when theminority channel has a gap after 12 filled states and the majorityhas a gap after 14 filled states. The multiple gap possibilitiesand the possibility of a magnetic semiconductor are illustratedby the partial band structures of Mn2CoAl shown in Fig. 2.Here, it can be seen that the down-spin channel has a gap after12 states per formula unit and the up-spin channel has potentialgaps after 9 and after 14 states per formula unit. The effect ofreduced symmetry on the potential up-spin gap after 9 statescan be seen in the avoided crossing of bands 9 and 10. For somesystems, this avoided crossing leaves only a tiny gap. A fullexplanation of possible band gaps requires an analysis of theband ordering at symmetry points and the symmetry-dependentband compatibility relations.

We note that other authors [7,8,12,34] use the term “Slater-Pauling gap” to refer to any gap that can create a half-metal ina Heusler compound. We prefer to reserve the term for gapsafter three states per atom. We apologize for any confusion ourattempt to refine nomenclature may cause.

The reduced-symmetry-derived gaps after 9 and 14 statesare often quite narrow so that a state called a spin gaplesssemiconductor (SGS), a special type of semiconductor in

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COMPUTATIONAL INVESTIGATION OF INVERSE … PHYSICAL REVIEW B 98, 094410 (2018)

which the Fermi level falls into a relatively large gap in theminority-spin channel (after 12 states) and within a narrow gapfor the majority (after 14 states) [34,35]. This band structureallows both electrons and holes to be excited simultaneouslycarrying 100% spin-polarized current, potentially with highmobility. Recently, the existence of some IH compounds hasbeen confirmed by experiment [36–38]. Additionally, it hasbeen experimentally verified that Mn2CoAl is an SGS withhigh Curie temperature [28].

Interestingly, the half Heusler C1b phase has the same spacegroup as the inverse Heusler XA phase and is also prone togaps after 9 and 14 states per formula unit in a spin channel.However, in this case 9 states per formula unit coincide withthe Slater-Pauling gap after 3 states/atom.

We anticipate that a large and complete database of con-sistently calculated properties of IH compounds will allow thetesting of hypotheses that may explain the occurrence and sizeof Slater-Pauling band gaps as well as the gaps that occur insome IH compounds after 9 and 14 states per formula unit.

Although a large number of IH compounds have beenanalyzed by first-principles calculations [12,34,39–42], a com-prehensive study of the structural, electronic, and magneticproperties of the IH family is useful because it is not clear howmany of the IH half-metals and semiconductors that can beimagined are thermodynamically stable in the XA structure.Thus, a systematic study of the structural stability of the IHfamily can provide guidance for future work.

In Sec. II we describe the details of our computationalapproach. Section II A describes the DFT calculations. InSec. II B, we discuss how the equilibrium structures for eachcompound were determined and the possibility of multiplesolutions in energy and magnetic configuration for somespecific compounds. Our approach to estimating the stabilityof the IH compounds is described in Sec. II C.

Our results are described in Secs. III and IV. Section IIIdescribes the formation energies that we calculated for theIH compounds and how these formation energies compare tothose of competing phases. Section III A describes a studyof the calculated formation energies and stability of thoseIH phases that have been observed experimentally. The re-lationship between stability and composition is discussed inSec. III B. Section III C describes our calculated results for theformation energy and stability of the 405 IH compounds thatwe studied, including an investigation of the relation betweengaps at the Fermi energy and stability. Section IV coversthe calculated electronic structure of the IH compounds withparticular emphasis on those that form semiconductors andhalf-metals. In Sec. IV A, IH semiconductors, half-metals, andnear-half-metals with negative formation energy are listed anddiscussed in terms of their electronic structure and structuralstability. Section V is a summary of our results and conclusions.

II. COMPUTATIONAL DETAILS AND STRUCTUREDETERMINATION

The IH compound X2YZ crystallizes in the face-centered-cubic XA structure with four formula units per cubic unitcell (Fig. 1). Its space group is no. 216, F 43m (the sameas the half Heusler). The IH structure can be viewed as fourinterpenetrating fcc sublattices, occupied by the X, Y , and Z

elements, respectively. The Z and Y elements are located at the(0, 0, 0) and ( 1

4 , 14 , 1

4 ), respectively, in Wyckoff coordinates,while the X1 and X2 elements are at ( 3

4 , 34 , 3

4 ) and ( 12 , 1

2 , 12 ),

respectively, resulting in two different rock-salt structures[X1Y ] and [X2Z] as shown in Fig. 1(e). We use X1 and X2 todistinguish these two X atoms sitting at the two inequivalentsites in the XA structure. Since we also calculated the energyfor each compound in the L21 structure, we also show the L21

structure in Fig. 1(c) for comparison.In this study, (a) X is one of 9 elements (Sc, Ti, V, Cr, Mn,

Fe, Co, Ni, or Cu), (b) Y is one of 9 elements (Ti, V, Cr, Mn,Fe, Co, Ni, Cu, or Zn), and (c) Z is one of 9 elements (Al, Ga,In, Si, Ge, Sn, P, As, or Sb). In summary, we calculated the 405inverse Heusler compounds for which the valence (or atomicnumber since X and Y are both assumed to be 3d) of Y is largerthan that of X. For each of these 405 potential IH compounds,we calculated its electronic and magnetic structure, stabilityagainst structural distortion, and formation energy. We alsocompared its formation energy to those of all phases andcombinations of phases in the open quantum materials database(OQMD) [43,44].

A. Density functional theory calculations

All calculations were performed using density functionaltheory (DFT) as implemented in the Vienna ab initio simulationpackage (VASP) [45] with a plane-wave basis set and projector-augmented wave (PAW) potentials [46]. The set of PAWpotentials for all elements and the plane-wave energy cutoff of520 eV for all calculations were both chosen for consistencywith the OQMD [43,44]. The Perdew-Burke-Ernzerhof (PBE)version of the generalized gradient approximation (GGA) tothe exchange-correlation functional of DFT was adopted [47].The integrations over the irreducible Brillouin zone (IBZ) usedthe automatic mesh generation scheme within VASP with themesh parameter (the number of k points per A−1 along eachreciprocal lattice vector) set to 50, which usually generated a15 × 15 × 15 �-centered Monkhorst-Pack grid [48], resultingin 288 k points in the IBZ. The integrations employed the lineartetrahedron method with Blöchl corrections [49]. Finally, allthe crystal structures are fully relaxed until the energy changein two successive ionic steps is less than 1 × 10−5 eV.

The total magnetic moment of each compound is calculatedby integrating the spin-polarized charge density over thecalculation cell. The division of the total magnetic momentamong the atoms in the cell is somewhat ambiguous in VASP.The magnetic moment for a particular atom presented in thispaper is evaluated within a sphere of 1.45 A radius centered onthe atomic site. It should be noted that the sum of the atomicmoments defined this way is not exactly equal to the total cellmoment.

B. Determination of the relaxed structure

We explain our procedure for obtaining the relaxed struc-tures in some detail in order to make clear that there are manypossible structures (in addition to the XA structure) that anX2YZ compound can assume and that the determination ofthe ground state can be complicated by the additional degreesof freedom associated with the possible formation of magnetic

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moments of varying magnitude and relative orientation and bypossible distortions of the cell.

We calculated the electronic structure of each compound inthe XA phase by minimizing the energy with respect to thelattice constant. For each system and each lattice constant, wefound it to be important to initiate the calculation with multipleinitial configurations of the magnetic moments. We consideredfour kinds of initial moment configurations: (1) moments onthe three transition-metal atoms are parallel; (2) moment onthe Y site is antiparallel to the moments on the X1 and X2;(3) moment on the X1 site is antiparallel with the moments onthe X2 site and Y ; (4) moment on the X2 site is antiparallel withthe moments on the X1 and Y . After the lattice constant andmagnetic configuration that determined the minimum energyfor the XA structure was determined, we compared this energyto the minimum energy determined in a similar way for the L21

structure.Competition between XA and L21. Comparing the calcu-

lated energies of the XA and L21 phases of the 405 compoundsthat we considered, we found that 277 have lower energy inthe L21 phase. The remaining 128 have lower energy in theXA phase. This result may be compared to the conclusionin Refs. [12,40,50–55] that the XA structure is energeticallypreferred to the L21 structure when the atomic number of theY element is larger than the atomic number of X. Althoughthis hypothesis is consistent with some experimental observa-tions [37,38,40,56,57], it does not seem to be generally validbecause more than two thirds of the systems in our data set(all of which satisfy NY > NX) have lower calculated energyas L21.

Test for tetragonal distortions via ionic relaxation. Afterdetermination of the cubic lattice constant and magneticconfiguration in our initial survey, all of the 405 compoundsin our survey, both L21 and XA, were tested for stabilityagainst tetragonal distortions using an 8-atom tetragonal celland full ionic relaxation. For both XA and L21 we consideredthe structure to be cubic if |c/a − 1| < 0.01. All relaxationsstarted from the XA structure or L21 structure adapted to an8-atom tetragonal cell.

Of the 277 compounds that preferred the L21 structure, 136remained in the L21 structure while 141 relaxed to a tetragonal“L21” structure. We notice that few systems with X = Sc, Ti, orV are calculated to be stable as XA. None of the 81 consideredcompounds with X = Sc were found to be XA. 10 relaxed intothe tetragonal “L21” phase, 63 preferred the L21 phase, andthe remaining 8 became tetragonal “XA” phase. Among the72 compounds with X = Ti, only 4 compounds preferred XA

phase, while 37 relaxed into the tetragonal “L21” phase, 29preferred the L21 phase, and 2 distorted to be tetragonal “XA”phase. Among the 63 compounds with X = V, 43 compoundsdistorted into tetragonal “L21” phase, 2 preferred the L21

phase, and 2 distorted to be tetragonal “XA” phase.When the 128 compounds that were more stable as XA than

L21 were tested for susceptibility to tetragonal distortion, 95were found to remain in the XA structure while 33 relaxed to atetragonal XA phase. In contrast to the half Heuslers which areprone to trigonal distortions [30], we find that the full Heuslersand IH compounds tend to retain tetragonal or cubic symmetryduring this type of relaxation. The susceptibility of the half

5.6 5.7 5.8 5.9 6 6.1 6.2-25

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rgy

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/f.u.

)

E=0.230 eV

L21 Ferrimagnetic1

L21 Ferrimagnetic2

L21 Nonmagnetic

XA Ferrimagnetic A2XA FerrimagneticXA Nonmagnetic

FIG. 3. Calculated total energies of Mn2ZnP with full-HeuslerL21 structure and inverse-Heusler XA structure as a function of thelattice constant a in ferrimagnetic, ferromagnetic and nonmagneticstates.

Heuslers to trigonal distortions is likely related to the vacantsite in the cell.

If we ignore the competition between XA and L21 phases,of the 405 potential XA compounds that we considered, 141were unstable with respect to a tetragonal distortion and theremaining 264 compounds have lower energy in XA phase.

Competition between magnetic configurations. For somecompounds, we found multiple solutions to the DFT equationsat the same or similar lattice constants with different magneticconfigurations. These can be categorized into six groups: inthe L21 structure, (1) the moments of the X atoms (which areidentical by assumption for L21) are parallel to those for Y

(L21 ferromagnetic), or (2) the moments of X are antiparallelto those for Y (L21 ferrimagnetic); in the XA structure, (3) themoments for Y are parallel with those for X1 but antiparallelwith those for X2 (XA ferrimagnetic A1), or (4) the momentsfor Y are parallel with those for X2 but antiparallel with thosefor X1 (XA ferrimagnetic A2), or (5) the moments for X1 andX2 are parallel with each other but antiparallel with those forY (XA ferrimagnetic), or (6) the moments for X1, X2, and Y

are parallel (XA ferromagnetic). Magnetic transitions, in bothmagnitude and orientation as a function of lattice constant, areclearly possible and often observed in our calculations. We didnot consider the possibility of noncollinear moments [58], andbecause the calculations did not include spin-orbit coupling, wedid not obtain information about the orientation of the momentsrelative to the crystal axes.

Example: Mn2ZnP. Mn2ZnP provides an example of com-petition between different magnetic states for both the L21 andXA structures as shown in Fig. 3. In this case, the L21 phaseshave lower energy. In addition to a nonmagnetic phase with alattice constant of 5.645 A, two energy minima were foundfor magnetic phases. These occur at a = 5.69 and 5.88 A.For the solution at a = 5.69 A the total moment per cell is2.53μB while the moments within a sphere of radius 1.45 Asurrounding each atom are 1.32 for Mn, −0.01 for Zn, and

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−0.10 for P. For the solution at a = 5.88 A the compoundhas a total magnetic moment of 5.37μB per unit cell, and themoments within the 1.45-A spheres in this case are 2.75μB ,−0.03μB , and −0.13μB for Mn, Zn, and P, respectively. Thesolution at a = 5.88 A has higher energy than the solution at5.69 A by �E = 0.230 eV per formula unit.

There are also multiple XA magnetic phases. The twoshown are both ferrimagnetic in the sense that the two Mnmoments have opposite signs. The phase with larger latticeconstant actually has smaller net moment because of the closerapproximate cancellation of the Mn moments.

To be clear, the energy comparisons of the precedingparagraph only consider the XA and L21 phases and theirpossible tetragonal distortions. Phase stability is discussedmore generally in Sec. II C.

Competition between ordered and disordered phases. Fora few systems, we considered the energy difference betweenordered and disordered phases. As will be discussed in detailin Sec. III A, some Mn2CoZ and Mn2RuZ XA compounds(especially when Z is a larger atom, e.g., In, Sn, or Sb,) seemto have disorder in the occupation of the Mn1 and Y sublatticesites. To approximate this disorder in the XA Mn2CoZ (Z =In, Sn, and Sb) and Mn2RuSn, special quasirandom structures(SQS) as described by Zunger et al. [59] were generated. TheSQS are designed to best represent the targeted disordered alloyby a periodic system of a given size.

Our SQS comprised 32-atom cells for the case of completelyrandom (50%/50%) occupation of the X1 and Y sites, and64-atom cells for the case in which the X1 (Y ) site occupationwas 75%/25% (25%/75%). The SQS was generated using theMonte Carlo algorithm of Van de Walle et al. [60] as imple-mented in the alloy theoretic automated toolkit (ATAT) [61].The cluster correlations used to define the SQS were specifiedusing a distance-based cutoff of all two-, three-, and four-bodyclusters with a maximum distance of 5–6 A. We did notconsider the case with smaller X/Y ratios because much largerunit cells would have been required. More detailed results willbe discussed in Sec. III A.

C. Calculation of energetic quantities

1. Formation energy

We further investigated the structural stability of the 405 po-tential IH compounds by calculating their formation energies.The formation energy of an IH compound X2YZ is defined as

�Ef (X2YZ) = E (X2YZ) − 14 (2μX + μY + μZ ), (1)

where E (X2YZ) is the total energy per atom of the IHcompound, and μi is the reference chemical potential ofelement i, chosen to be consistent with those used in theOQMD database. Typically, the reference energy is the totalenergy per atom in the elemental phase (see Refs. [43,44]for details). A negative value of �Ef indicates that at zerotemperature, the compound is more stable than its constituentelements. It is a necessary but not sufficient condition forground-state thermodynamic stability. It does not, for example,guarantee the stability of an inverse Heusler (IH) phase overanother competing phase or mixture of phases. For that, wemust determine the convex hull.

2. Convex hull distance

A compound can be thermodynamically stable only if itlies on the convex hull of formation energies of all phasesin the respective chemical space. Every phase on the convexhull has a formation energy lower than any other phase orlinear combination of phases in the chemical space at thatcomposition. Thus, any phase on the convex hull is, bydefinition, thermodynamically stable at 0 K. Conversely, anyphase that does not lie on the convex hull is thermodynamicallyunstable; i.e., there is another phase or combination of phaseson the convex hull that is lower in energy.

The distance from the convex hull �EHD for a phase withformation energy �Ef can be calculated as

�EHD = �Ef − Ehull, (2)

where Ehull is the energy of the convex hull at the compositionof the phase. The energy of the convex hull at any compositionis given by a linear combination of energies of stable phases.Thus, the determination of Ehull from a database of formationenergies is a linear composition-constrained energy minimiza-tion problem [62,63], and is available as a lookup featurecalled “grand canonical linear programming”(GCLP) on theOQMD website (http://oqmd.org/analysis/gclp). Obviously, ifthe hull distance �EHD for a phase on the convex hull is 0,there is no other phase or linear combination of phases lowerin energy than the phase at that composition. The distance ofthe formation energy of a phase from the convex hull is anindicator of its thermodynamic stability and the likelihood ofit being fabricated because the greater the distance from theconvex hull, the higher is the thermodynamic driving forcefor transformation into another phase or decomposition into acombination of phases.

We note that the distance of a phase from the calculatedconvex hull depends on the completeness of the set of phasesconsidered in the construction of the convex hull. Ideally, tocalculate the convex hull of a system, X-Y -Z, one wouldcalculate the energies of all possible compounds that can beformed from elements X, Y , and Z. Unfortunately, such acomprehensive study is not currently feasible.

A practical approach is to construct the convex hull usingall the currently reported compounds in the X-Y -Z phasespace. Here, we have limited our set of considered phases tothose in the OQMD, which includes many ternary phases thathave been reported in the inorganic crystal structure database(ICSD) [64,65], and ∼500 000 hypothetical compounds basedon common crystal structures. Thus, the calculated formationenergy of each X2YZ IH compound is compared againstthe calculated formation energies of all phases and all linearcombinations of phases with total composition X2YZ in theOQMD.

A phase that we calculate to be above the convex hull maystill be realized experimentally for three reasons: (a) Theremay be errors in the calculation of the energies of the IHcompound or in the reference compounds. These errors maybe intrinsic to PBE-DFT or due to insufficient precision in thecalculation or inability to distinguish the ground state. We haveapplied considerable effort to reduce or eliminate the lattertwo possibilities so far as possible. (b) The compounds areusually synthesized at temperatures well above 0 K. Because

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http://oqmd.org/analysis/gclp


entropic effects will differ for different phases, the phasewith lowest free energy at the synthesis temperature may bedifferent from the phase with lowest energy at 0 K and thishigh-temperature phase may persist to low temperature dueto slow kinetics at laboratory time scales. (c) Nonequilibriumprocessing techniques can sometimes be used to fabricatenonequilibrium metastable phases.

Conversely, an IH phase that we calculate to be on the calcu-lated convex hull may nevertheless be difficult or impossible tofabricate because of reasons (a) or (b) or because the databaseused to generate the convex hull does not contain all phases sothat the true convex hull lies below the calculated one.

Finally, we remark that our stability investigations are mostdirectly pertinent to bulk phases. The growth of alloy films is asophisticated art. Phases that are stable in bulk can be difficultto grow as films. Conversely, clever use of substrates, surfaceenergies, and kinetics can stabilize film structures differentfrom the equilibrium bulk phase. Even for films, however,the hull distances provided here may provide useful guidanceconcerning the likelihood that a phase or structure may bestabilized.

III. FORMATION ENERGY AND STABILITY

A. Formation energy and convex hull distance

In this section, we systematically investigate the formationenergy and thermodynamic stability of the 405 IH compoundsconsidered in this paper. The formation energies at 0 K wereevaluated by using Eq. (1) and the convex hull distances werecalculated using Eq. (2). We first explored the relationshipbetween formation energy and convex hull distance for theknown synthesized IH compounds, including those collectedin the ICSD.

We compiled a list of the reported IH compounds by ex-tracting IH compounds from the ICSD and literature includingall elements as potential X, Y , or Z atoms. From this listwe removed the compounds with partially occupied sites.Some of the ICSD entries were from DFT calculations. Thesewere included except for those with a higher DFT energythan the corresponding full Heusler which were obviouslyincluded in the ICSD by mistake. Finally, we tabulated theformation energy and convex hull distance as calculated in theOQMD.

A total of 48 distinct IH compounds are reported in ICSD.Of these, 36 have been synthesized experimentally. Referencesto six additional synthesized IH compounds, not included inICSD (Fe2CoGa, Fe2CoSi, Fe2CoGe, Fe2NiGa, Mn2NiGa,and Fe2RuSi) were found in recent literature [66,67]. These canbe used for cross validation of our method. Although Mn2NiSnwas synthesized by Luo et al. [68], a more recent study showsthat it has atomic disorder on the transition-metal sites [67]and our DFT calculation shows it is 103 meV/atom above theconvex hull.

These data from ICSD and the recent literature are displayedin Fig. 4, from which it can be seen that most of the synthesizedIH compounds are within 52 meV/atom of the convex hull.The green diamonds represent the IH compounds that havenot been synthesized but have been sourced into the ICSD byDFT predictions.

FIG. 4. DFT-calculated formation energy vs convex hull distanceof inverse Heuslers reported in ICSD and in recent experimentalstudies. A hull distance �EHD = 0 (on the convex hull) indicatesthe compound is thermodynamically stable at 0 K. Blue circlesand red squares represent experimentally synthesized IH throughequilibrium and nonequilibrium process, respectively. Purple triangleis synthesized IH coexisting with other phase. Yellow stars representsynthesized compounds reported in literature but not in ICSD. Greendiamonds represent IH compounds predicted using DFT.

Overall, we find most of the experimentally reported IHcompounds (38 of 42) have a convex hull distance less than52 meV/atom. Thus, the convex hull distance of the DFT-calculated formation energy at 0 K appears to be a goodindicator of the likelihood of synthesis of an IH compoundand 52 meV/atom is a reasonable empirical threshold sepa-rating the compounds likely to be synthesized by equilibriumprocessing from those unlikely to be so synthesized.

The empirical 52-meV threshold assumed here may be con-trasted with the 100-meV threshold that seemed appropriatefor the half Heusler compounds [30]. We speculate that themore open structure of the half Heusler phase is associatedwith a high density of states for low-frequency phonons whichreduces the free energy of the half Heusler phase relative tocompeting phases at temperatures used for synthesis.

Before proceeding, it is important to note that severalMn2YZ compounds (Mn2CoSn, Mn2CoSb, Mn2CoIn, andMn2RuSn) appear to significantly violate our empirical hulldistance threshold. They also have vanishing or positive for-mation energy! A discussion of our rationale for not extendingour hull distance threshold to include these systems follows.

(a) Mn2CoSn: Our calculations predict that XA phaseMn2CoSn lies well above the convex hull. A mixture of phases,CoSn-Mn is predicted to be lower in energy by �EHD =0.107 eV/atom. However, there are reasons to believe that theexperimentally reported phase may not be pure XA.

The ICSD entry for Mn2CoSn is based on the work ofLiu et al. who described synthesis via melt spinning as wellas calculations that employed the FLAPW technique [40].They found that the nonequilibrium melt-spinning techniquegenerated a material with a “clean” x-ray diffraction (XRD)pattern consistent with the XA phase. The measured magnetic

094410-6


moment of 2.98μB per formula unit was consistent withtheir calculated value of 2.99 and with their calculated DOS,which indicates that this material may be a Slater-Paulingnear-half-metal.

Subsequently, Winterlik et al. synthesized this compoundby the quenching from 1073 K of annealed arc-melted sam-ples [38]. They obtained similar XRD results to Liu et al., butwith a small admixture of a tetragonal MnSn2 phase. Theyalso concluded that the experimental moment was near theSlater-Pauling half-metal value of 3μB per formula unit. How-ever, their calculations for XA-phase Mn2CoSn which alsoemployed FLAPW found, in addition to the near-half-metallicphase found by Liu et al., a different magnetic structure withlower energy.

Our calculations using VASP confirm the Winterlik et al.results. The PBE-DFT predicted phase for XA Mn2CoSnhas a magnetic moment around 1.53μB rather far from thevalue of 3μB necessary to place the Fermi energy into theSlater-Pauling gap after 12 states per atom in the minoritychannel. There is an interesting competition between twovery similar ferrimagnetic states that plays out in many ofthe 26- and 27-electron Mn2CoZ systems. The Mn2 and Comoments are aligned and are partially compensated by the Mn1

moment. For smaller lattice constants, the compensation yieldsthe correct moment for a half-metal or near-half-metal [i.e.,for Mn2Co(Al,Ga,Si,Ge)], but as the lattice constant increases[i.e., for Mn2Co(In,Sn)], the Mn1 moment becomes too largein magnitude and the partial compensation fails to yield ahalf-metal.

Winterlik et al. conclude from XRD, NMR and Mössbauerstudies that their quenched samples of Mn2CoSn are actuallydisordered on the MnCo layers. They modeled this disorderusing the Korringa-Kohn-Rostoker (KKR) and the coherentpotential approximation (CPA) (KKR-CPA) approach andobtained values of the moments in reasonable agreementwith magnetometry and x-ray magnetic circular dichroism(XMCD). They argued that their model with Mn and Co ran-domly occupying the sites on alternate (001) layers separatedby ordered Mn-Sn layers agreed with XRD as well as theXA model because Mn and Co scatter x rays similarly. Thisdisordered L21B structure has both Mn and Co at the ( 1

4 , 14 , 1

4 )and the ( 3

4 , 34 , 3

4 ) sites, whereas the XA structure has only Coat ( 1

4 , 14 , 1

4 ) and only Mn at ( 34 , 3

4 , 34 ) site.

We investigated the Winterlik et al. hypothesis by com-paring the energy of a few cubic 16-atom supercells withdiffering equiatomic occupations of the MnCo layers to theenergy of the XA occupation. The energies of these differentsite occupations were all lower than the XA energy, andmost of them were half-metallic or near-half-metallic. Thedecrease in energy for the non-XA site occupations rangedfrom 8 to 33 meV/atom. These should be considered lowerbound estimates of the decrease since we did not optimizethe geometries of the model “disordered” structures. We alsoperformed calculations for an SQS designed to mimic a systemwith disorder on the MnCo layers. Our calculation indicatesthat the SQS with 25% Mn/Co mixing is 31 meV/atom lowerin energy than the XA phase. The net magnetic moment of thisSQS is about 1.28μB per f.u., which is lower than that of XA

phase (1.53μB) due to the atom disordering.

We propose the following hypothesis and rationale foromitting Mn2CoSn from the estimate for the hull energydistance threshold: During synthesis at high temperature,Mn2CoSn is thermodynamically stable in a phase with sub-stantial configurational disorder on the MnCo planes. Theconfigurational disorder on these sites may also decrease thefree energy of the compound. A simple mean-field estimate ofthe configurational entropy (S) of disorder gives a −T S con-tribution of −32 meV/atom at 1073 K. If we add this entropiccontribution to the approximately 30-meV/atom difference intotal energy between the ordered and disordered systems, wecan estimate a decrease in the relative free energy of formationby approximately 60 meV/atom due to disorder. This wouldyield a negative free energy and a hull distance less than ourempirical 52-meV threshold.

(b) Mn2CoSb: Mn2CoSb is predicted to lie above the convexhull with a mixture of phases CoMnSb-Mn lower in energy by�EHD = 0.133 eV/atom. This phase also entered the ICSDdatabase from the paper of Liu et al. who generated theirsample by melt spinning and also provided FLAPW calculationsthat predicted XA Mn2CoSb to be a half-metal with a Slater-Pauling moment of 4μB per formula unit, consistent with theirmeasured value.

Unlike some Mn2CoZ compounds with smaller Z atoms,synthesis of single XA phase Mn2CoSb is difficult via arcmelting and a high-temperature annealing step. Prior x-raydiffraction investigations [40] revealed a multiphase Mn2CoSbin arc-melt samples. Our synthesis supports this finding [69].On the other hand, a melt-spinning treatment of arc-meltMn2CoSb ingots is reported to produce a pure cubic struc-ture [40,70].

We consider it likely that Mn2CoSb is similar to Mn2CoSn.A study by Xin et al. found that “disordered” Mn2CoSb [appar-ently modeled by replacing the (001) MnCo layers by layersof pure Mn and pure Co alternating with the MnSb layers]was lower in energy than XA by 21 meV/atom [39]. Our SQScalculations show that the 50% Mn/Co mixture within onerock-salt lattice is 60 meV/atom lower in energy than the XA

phases. Again, our assertion is that the free-energy reductiondue to configurational entropy together with the lower energyof the disordered configurations may be sufficient to stabilizethe L21B phase as it is rapidly quenched from the melt.

(c) Mn2CoIn: According to Liu et al. [40], Mn2CoInappears to be multiphase in samples prepared by arc melting.Apart from the cubic phase identified as inverse Heusler, asecond phase described as fcc is also apparent. Liu et al.also report FLAPW calculations performed at a much smallerlattice constant (5.8 A) than the experimentally derived latticeconstant (6.14 A). These calculations yielded a moment of1.95μB in reasonable agreement with experimental value of1.90μB . A subsequent calculation by Meinert et al. found alattice constant of 6.04 A and a moment of 1.95μB [52].

Our calculations for the XA phase yielded a lattice constantof 6.10 A, much closer to the experimental value and a momentof 1.74μB . We also found a surprising and unusual sensitivityof our results to the number of plane waves. We performedSQS calculations which indicated that the disordered phaseis essentially degenerate in energy with the XA phase. Themagnetic moment of SQS is about 1.01μB per f.u.

094410-7


Given the significant amount of second phase in the experi-mental sample which would affect the composition of the phaseidentified as XA, together with the possibility of disorder onthe MnCo layers we tentatively conclude that it is reasonableto discount the large calculated formation energy and hulldistance in setting our threshold for likely fabricability of XA

phases. Additional studies of this putative phase are probablyadvisable.

(d) Mn2RuSn: Mn2RuSn is predicted to lie above theconvex hull with a mixture of phases MnSnRu+Mn+Sn7Ru3

by �EHD = 0.085 eV/atom. The first reports of fabricationby Endo et al. [71], were unable to distinguish between theXA and L21B phases. A later study by Kreiner et al. seemedto confidently conclude from powder XRD studies that theirsample prepared by alloying through inductive heating had theL21B structure [67]. Our calculation of the energy of a SQSfor Mn2RuSn with 25% Mn/Ru mixing gave an energy of themodel disordered alloy 18 meV lower than XA. The calculatedmagnetic moment of the SQS is about 0.24μB per f.u.. A studyby Yang et al. using the KKR-CPA approach estimated 11%antisite Mn based on a comparison of the calculated momentto the experimental one [72].

We tentatively conclude that the four experimental systemsin the ICSD with anomalously large calculated hull distancesand formation energies are probably not XA phase and excludethem from consideration in determining the threshold forestimating which XA phases are likely to be fabricated by equi-librium approaches. We also remind that an additional potentialinverse Heusler, Mn2NiSn, was excluded from our database ofexperimental XA phases, because of experimental evidence ofsimilar disorder to that observed in the Mn2CoZ phases.

We recommend additional studies of the competition be-tween XA and L21B phases, especially for compounds whichhave large-Z atoms. Such studies may require careful experi-mental analysis because of the similarities of the XRD patternsof the two phases. Theoretical studies may also be difficultbecause of the need to accurately calculate the formationenergy of the disordered L21B phase.

We extended the comparison of formation energy andconvex hull distance to all the 405 IH compounds consideredin this work as shown in Fig. 5. It can be seen from Fig. 5 thatthe calculated formation energies span a range from −0.45 to0.38 eV/atom. This may be compared with a range of −1.1to 0.7 eV/atom that was observed for half Heuslers [30]. 248of the IH have negative formation energy, indicating thermalstability against decomposition into the constituent elementsat 0 K. The calculated hull distances vary from 0 to nearly1 eV/atom. It is clear from the lack of correlation betweenformation energy and hull distance in Fig. 5 that the formationenergy is not a reliable indicator of stability.

The IH compounds considered in this work that are includedin ICSD are indicated by red squares in Fig. 5. These all liewithin 0.052 eV/atom of the convex hull except for threecompounds mentioned previously (Mn2CoSn, Mn2CoIn, andMn2CoSb) that were synthesized by nonequilibrium process-ing and are likely L21B rather than XA phase. Our calcu-lations predict 13 (out of 405) IH compounds to be within52 meV/atom of the convex hull. The “success rate,” i.e., thenumber of predicted potentially stable compounds versus totalnumber of systems investigated is lower for IH (13/405) than

FIG. 5. DFT-calculated formation energy vs convex hull distanceof all IH considered in this paper. A hull distance �EHD = 0 (onthe convex hull) indicates the compound is thermodynamically stableat 0 K. Red diamonds and green squares represent IH compoundssourced into ICSD by synthesis and DFT computation, respectively.Blue circles represent the IH compounds calculated in this work.

for half Heusler (50/378) [30], but this result is affected by thedifferent hull distance thresholds and the different choices ofpotential compounds.

B. Stability and composition

We systematically investigated the formation energies ofthe 405 IH compounds in the XA phase. We found that manyof these potential IH phases actually have lower energy inother phases, e.g., L21(136), tetragonal “L21” phase (141),or tetragonal XA(33). Here, however, since we are interestedin gaining insight into the systematics of the stability of the XA

phase, we compare formation energies as if all 405 compoundsremained in the XA phase. To better illustrate the relationbetween structural stability and composition, we arranged theformation energy data according to X, Y , and Z elements,respectively, in Figs. 6–8.

Figure 6 shows the calculated formation energies and hulldistances of the 405 calculated X2YZ compounds arrangedby X species, then by Y , and finally by Z. From Fig. 6, wecan see that Sc, Ti, and V generate more negative formationenergy XA compounds as X elements. In terms of the fractionwith negative formation energy, we found 75/81 with X = Sc,60/72 with X = Ti, 25/45 with X = Mn, and 20/36 with X =Fe, and 15/18 with X = Ni.

Figure 7 presents the calculated formation energies and hulldistances arranged first by Y , then by X, and finally by Z. FromFig. 7, it can be seen that when Y is Fe, Co, or Ni many XA

compounds with negative formation energies result. However,when one considers the fraction of negative formation energycompounds with a given Y in our database, the highest are 9/9for Y = Ti, 17/18 for Y = V, and 24/27 for Y = Cr.

When the compounds are ordered by Z element as presentedin Fig. 8, we can see that the smaller atoms within a given group(i.e., Z = Al, Si, In) tend to have a larger fraction of negative

094410-8


FIG. 6. DFT formation energies and hull distances for potentialinverse Heusler compounds grouped by the element on the X site.The numbers near the top (in blue) and center (in brown) of eachcolumn denote the number of compounds with negative formationenergy �Ef and hull distance �EHD � 0.1 eV/atom, respectively,in the corresponding Z-element group. Within a given X-elementcolumn, the compounds are ordered first by the element on the Y site(same order as in Fig. 7) and then by the element on the Z site (sameorder as in Fig. 8), i.e., Z varies more rapidly than Y .

energy XA phases. The large number of �Ef < 0 compoundswith Z = Al is striking in Fig. 8, but only 14 compounds havenegative formation energy for Z = Sb. Although the number ofnegative formation energy compounds with Z = Al is greatest,the number among them with formation energies less than−0.2 eV (13) is less than for compounds with Z = Si (25)and Z = P (24).

FIG. 7. DFT formation energies and hull distances for potentialinverse Heusler compounds grouped by the element on the Y site. Thenumbers near the top (in blue) and center (in brown) of each columndenote the number of compounds with negative formation energy�Ef and hull distance �EHD � 0.1 eV/atom, respectively, in thecorresponding Y -element group. Within a given Y -element column,the compounds are ordered first by the element on the X site (sameorder as in Fig. 6) and then by the element on the Z site (same orderas in Fig. 8), i.e., Z varies more rapidly than X.

FIG. 8. DFT formation energies and hull distances for potentialinverse Heusler compounds grouped by the element on the Z site. Thenumbers near the top (in blue) and center (in brown) of each columndenote the number of compounds with negative formation energy�Ef and hull distance �EHD � 0.1 eV/atom, respectively, in thecorresponding Z-element group. Within a given Z-element column,the compounds are ordered first by the element on the X site (sameorder as in Fig. 6) and then by the element on the Y site (same orderas in Fig. 7), i.e., Y varies more rapidly than X.

It is interesting to contrast the trends in formation energyto those observed by considering the OQMD hull distance.Although Sc and Ti tend to generate negative formation ener-gies as X atoms in XA phase X2YZ compounds, the picture isopposite for the hull distances. Very few of these compoundshave a hull distance near the empirical 0.052 eV/atom rangewhich is indicative of a potentially synthesizable phase. Usinga larger hull distance of 0.1 eV as an (arbitrary) stabilitythreshold, we find no X = Sc phases that meet the criterion andonly 2 of 72 X = Ti phases. Sc and Ti are “active” elementsthat form numerous low-energy phases. Hull distances tend tobe lower when the X atom is Mn or Fe with 22 of 45 X = Mnand 12 of 36 X = Fe meeting the criterion of hull distance lessthan 0.1 eV/atom.

While 9/9 Y = Ti and 17/18 Y = V IH compounds havenegative formation energy, none of them meet the relaxedhull distance threshold of 0.1 eV/atom. However, relativelylarge fractions of Y = Fe (10/45), Y = Co (11/45), and Y =Ni (8/63) meet the criterion. Thus, there is better correlationbetween low formation energy and low hull distance when theXA compounds are grouped by Y element. By both measures,Y = Ni, Co, or Fe tend to generate more stable compounds.

The trends in stability with Z atom are also somewhatsimilar when viewed from the perspectives of hull distance andformation energy. Al, Ga, Si, and Ge as Z atom are associatedwith compounds having relatively low hull distances andnegative formation energies. One big difference, however, isZ = P which tends to generate numerous XA compounds withlow formation energy, but none with very low hull distances.

C. Stability and gaps near the Fermi energy

We calculated the electronic structure of each compoundand obtained its spin polarization at the Fermi level EF . The

094410-9


FIG. 9. The distribution of the 405 potential inverse Heuslercompounds with negative (�Ef < 0 eV/atom) and positive (�Ef >

0 eV/atom) formation energies as a function of spin polarizationP (EF ) [given by Eq. (3)]. In the central region, we show the numberof inverse Heusler compounds grouped by 10 percentage points ofspin polarization. In an additional region to the left, we show the 15semiconductors, including 14 compounds with a negative formationenergy and 1 with a positive formation energy. In the additional regionon the right, we show 52 half-metals, including 51 with negativeformation energy and 1 with positive formation energy.

spin polarization P at EF is defined as

P (EF ) = N↑(EF ) − N↓(EF )

N↑(EF ) + N↓(EF ), (3)

where N↑ and N↓ are the densities of states for majority(spin-up) and minority (spin-down) electrons, respectively. Wedefine the compounds with 100% spin polarization [P (EF ) =1] to be half-metals in this paper.

In a study of the half-Heusler compounds, gaps at the Fermienergy in one or both spin channels appeared to be associatedwith low formation energies and low hull distances [30].Figure 9 shows how the number of IH compounds with positiveand negative formation energy varies with spin polarization. Itcan be seen that for semiconductors and half-metals the ratioof the number of negative formation energy compounds topositive formation energy compounds is particularly high.

Figure 10 shows the number of IH compounds with pos-itive and negative hull distances less than and greater than0.1 eV/atom versus spin polarization. In contrast to the caseof the half Heuslers, the plot of hull distance versus spinpolarization does not present an obvious case for a correlationbetween gaps near the Fermi energy and stability. As willbe discussed in Sec. IV A, this is due to the large numberof X = Sc and X = Ti compounds in our database that aresemi-conductors or half-metals and although they tend to havenegative formation energies, they tend to be less stable thancompeting compounds. On the other hand, as will be discussedin Sec. III, only 20 of the IH compounds in our database sys-tems met our 0.052-eV/atom hull distance stability criterion.Half of these are half-metals or near-half-metals. We considerthis to be strong support for the notion that gaps in or near

FIG. 10. The distribution of the 405 potential inverse Heuslercompounds that lie near (�EHD < 0.1 eV/atom) or far (�EHD >

0.1 eV/atom) from the convex hull, as a function of spin polarizationP (EF ) [given by Eq. (3)]. In the central region, the number ofinverse Heusler compounds grouped by 10 percentage points of spinpolarization is shown. In the additional region on the left are shown 15semiconductors (of which 2 compounds have EHD < 0.1 eV/atom).On the right are shown the 52 half-metals (of which 9 compoundshave EHD < 0.1 eV/atom).

the Fermi energy in one or both spin channels contribute tostability.

IV. ELECTRONIC STRUCTURE

In this section the electronic structures of the IH compoundsare discussed with a particular emphasis on gaps in the densityof states, particularly those after 9, 12, and 14 states per formulaunit.

A. IH semiconductors and half-metallic ferromagnets

In total we found 14 semiconductors, 51 half-metals, and50 near-half-metals with negative formation energy among the405 potential IH compounds. We will discuss these materialsin this section.

1. Semiconductors

The 14 XA semiconductors with negative formation energythat we found are listed in Table I where the DFT-calculatedproperties (i.e., lattice constant, band gap, gap type, localmoments, formation energy, hull distance) are presented.Remarkably, all of these 14 systems are 18 electron non-Slater-Pauling semiconductors with gaps after 9 electrons performula unit (2.25 per atom) per spin channel. Recall thatSlater-Pauling gaps (at least according to the nomenclaturewe prefer) occur after 3 electrons per atom per spin.

From Table I, it can be seen that all of these semiconductorshave either Sc or Ti as X atoms. This is not surprising (giventhat they all have 18 electrons per formula unit) because thechoice of systems that comprise our database (see Sec. II)implies that if the number of valence electrons on an X atom

094410-10


TABLE I. For each of the 14 18-electron X2YZ inverse Heusler compounds that are non-Slater-Pauling semiconductors with negativeformation energy, we list the calculated lattice constant a, band gap in two spin channels E↑

g , E↓g within DFT, local moments for atoms on the

X, Y , and Z sites: m(X1), m(X2), m(Y ), and m(Z), the formation energy of the compound in the XA and L21 structures, distance from theconvex hull for the XA phase �EXA

HD , previous experimental reports.

a E↑g E↓

gm(X2) m(Y ) �EXA

f �EL21f �EXA

HD

X2YZ (A) (eV) (eV) m(X1) (μB ) m(Z) (eV/atom)

Sc2MnP 6.18 0.052 0.052 0 0 0 0 − 0.286 − 0.471 0.631Sc2FeSi 6.255 0.347 0.347 0 0 0 0 − 0.292 − 0.416 0.311Sc2FeGe 6.33 0.356 0.356 0 0 0 0 − 0.262 − 0.435 0.371Sc2FeSn 6.60 0.337 0.337 0 0 0 0 − 0.230 − 0.358 0.319Sc2CoAl 6.42 0.475 0.475 0 0 0 0 − 0.280 − 0.350 0.130Sc2CoGa 6.36 0.532 0.532 0 0 0 0 − 0.328 − 0.454 0.159Sc2CoIn 6.61 0.470 0.470 0 0 0 0 − 0.205 − 0.281 0.200Ti2VSb 6.47 0.427 0.119 − 1.236 − 0.521 1.881 − 0.009 − 0.215 − 0.208 0.162Ti2CrSi 6.11 0.475 0.086 − 1.157 − 0.787 2.129 − 0.036 − 0.337 − 0.385 0.230Ti2CrGe 6.20 0.522 0.104 − 1.254 − 0.883 2.348 − 0.022 − 0.262 − 0.305 0.259Ti2CrSn 6.47 0.579 0.133 − 1.396 − 1.018 2.727 − 0.016 − 0.181 − 0.154 0.139Ti2MnAl 6.23 0.561 0.043 − 1.202 − 1.067 2.606 − 0.043 − 0.268 − 0.302 0.088Ti2MnGa 6.20 0.603 0.034 − 1.201 − 1.115 2.611 − 0.020 − 0.287 − 0.341 0.114Ti2MnIn 6.47 0.394 0.042 − 1.336 − 1.266 3.043 − 0.020 − 0.100 − 0.05 0.092

exceeds 4, the total number of electrons per formula unit willexceed 18.

Of the 9 XA compounds with X = Sc in our database, 7 arecalculated to be nonmagnetic semiconductors. The two that arenot semiconductors are Sc2MnAs and Sc2MnSb. The formeris nonmagnetic with a very low DOS pseudogap at the Fermienergy.

The absence of a gap in Sc2MnAs can be traced to areordering of the states at �. Between Sc2MnP and Sc2MnAs, asingly degenerateA1 state (band 7) switches places with a triplydegenerate T2 state (bands 8, 9, and 10), as shown in Fig. 11.This A1 state (composed of s contributions from all fouratoms) decreases in energy relative to the d states as the latticeexpands from Sc2MnP to Sc2MnAs to Sc2MnSb. However,

-2.5

-2

-1.5

-1

-0.5

0

0.5

1(b) Sc

2MnAs

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

E -

Ef (

eV)

(a) Sc2MnP

10

8 899

7 7

10T

2

A1

K KL L

FIG. 11. The band structures of (a) Sc2MnP and (b) Sc2MnAs areshown. The singly degenerate A1 state and triply degenerate T2 stateare labeled in (b). Bands 7, 8, 9, and 10 are labeled in (a) and (b). Thezero of the energy axis corresponds to the Fermi level.

Sc2MnSb becomes magnetic as the lattice is expanded beforethis reordering occurs so it is predicted to be a magneticnear-half-metal with a tiny moment (see Table III).

The reason Sc2MnSb is magnetic while Sc2MnP andSc2MnAs are not is almost certainly due to the larger latticeconstant induced by the larger atom (Sb). For lattice constantsless than approximately 6.50 A, Sc2MnSb is predicted to be anonmagnetic semiconductor similar to Sc2MnP.

The DOS of the nonmagnetic Sc-based semiconductors inTable I can be understood by referring to Fig. 12(a) whichshows the site decomposed DOS for Sc2CoAl. Because Sc isan early transition metal compared to Co, its d states must liehigher. Thus, the ordering of the states is Al-s (in this caseforming a narrow band well below the Fermi level and notshown in the figure.) followed by Al-p, followed by Co-d, andfinally Sc1-d and Sc2-d. The gap after 9 electrons per spinchannel occurs from the hybridization of the Co-d and Sc-dstates or, more generally, between the Y -d and X-d states.

Thus, bonding in these systems appears to have a consider-able ionic component with transfer of electrons to the Al andCo atoms from the Sc atoms. It can be seen from Table I thatall of the X = Sc 18-electron semiconductors are predictedto be substantially more stable in the L21 phase than in theXA phase. In addition, the XA hull distances all exceed our0.052-eV threshold for potentially being stable in equilibrium.

Of the 9 18-electron compounds in our database with X =Ti, 7 are predicted to be semiconductors and are included inTable I. One, Ti2VP, is predicted to be a near-half-metal (seeTable III) with a tiny moment. The final compound, Ti2VAs, ispredicted to be a magnetic semiconductor similar to Ti2VSb,but its formation energy is greater than zero and so is omittedfrom Table I.

In contrast to the nonmagnetic 18-electron semiconductorswith X = Sc, the systems with X = Ti develop magneticmoments and have gaps of different size in the majority-and minority-spin channels. The net moment per formula unitis predicted to be zero for the semiconductors, with the Ti

094410-11


FIG. 12. Density of states curves for the 18-electron semicon-ductor (a) Sc2CoAl and (b) Ti2CrSn. The curve presents the totalDOS (black) and the projected DOS contributed by Sc1 (Ti1) (red),Sc2 (Ti2) (orange), Co (Cr) (blue), and Al (Sn) (green). Zero energycorresponds to the Fermi level.

moments having the same sign and being balanced by a largermoment of opposite sign on the Y site. If the sign of the momenton the Y atom is taken to be positive, then the larger gaps are inthe majority channel while the smaller gaps are in the minoritychannel, becoming slightly negative for Ti2VP.

The larger gap (in the majority channel in Table I) for theTi2YZ magnetic semiconductors is very similar to the gapsin both channels for the Sc2YZ nonmagnetic semiconductors.The positive moment on the Y atom shifts its majority d leveldown while negative moments on the Ti atoms shift theirmajority d levels up, the larger moment on the Ti1 givingit a slightly larger upward shift. As a consequence, the stateordering in the majority channel is Z-s, Z-p, Y -d, Ti2-d, Ti1-dso that the majority gap after nine states per formula unitarises from hybridization between Y -d and X2-d and X1-dsimilar to the case for both channels in the Sc2YZ 18-electronsemiconductors.

The persistence of the very small gaps after nine states performula unit in the minority channel is surprising and seemsto be aided by avoided band crossings along � to K and alongL to W . We find these small gaps to be common when the d

levels of the three transition metals are similar. The avoidedcrossings and hence the gaps are likely symmetry related sincethe nonmagnetic compound V2VGa has several allowed bandcrossings that are avoided when the symmetry between X1,X2, and Y atoms is broken. These avoided band crossingslead to the small gap after nine states and are quite commonwhen X1, X2, and Y are similar, but not identical in one of thespin channels. For example, simple electron counting ignoringcharge transfer and using the calculated moments yields forTi2MnGa that the number of electrons for the minority channel

on each atom are approximately 2.6 for X1 and X2 and 2.2 forY . The corresponding minority gap is tiny, 0.034 eV.

The atom-projected DOS of Ti2CrSn is presented in Fig. 12.The energy ordering of the atomic orbitals is Sn-s (in thiscase forming a narrow band more than 10 eV below the Fermilevel and not shown in the figure.) followed by Sn-p, followedin the majority channel by Cr-d, and then the closely spacedTi2-d and Ti1-d. In the minority channel the ordering of thed states is different, Ti1-d, Ti2-d, followed by Cr-d. In themajority channel, the hybridization between the lower-lyingCr-d and the higher Ti-d states produces a relatively large gap(0.579 eV), while in the minority channel, the relatively closelyspaced Cr-d and Ti-d states create a smaller gap (0.133 eV).

Ti2CrSn might have interesting spin-dependent transportand tunneling properties if it can be synthesized. The calculatedhull distance for the compound is 0.139 eV/atom meaning thatthe OQMD contains phases or a combination of phases (in thiscase a mixture of binaries) that is lower in energy than our0.052-eV/atom empirical threshold.

We found no 24-electron Slater-Pauling semiconductorsamong the XA compounds that we investigated. Of the 405XA compounds that we studied, 24 had a valence electroncount of 24. None were semiconducting, though 6 were half-metallic with zero moment or near-half-metallic with a verysmall moment. The reason for the absence of 24-electronXA semiconductors is relatively simple. As we shall discussin the next section, the Slater-Pauling gap in the XA phaserequires formation of significant magnetic moments to createa significant difference between the d-onsite energies of thetwo X atoms. When moments form, the two spin channels willgenerally be different and a semiconductor becomes unlikely.The interesting exception to this rule is the 18-electron semi-conductors with moments, but zero net moment in which evenvery small differences between the d-onsite energies of the X

atoms can effectuate a small gap. All of the semiconductors inTable I have hull distances that exceed our stability thresholdcriterion of 0.052 eV/atom.

2. Half-metals and “near”-half-metals

Although a large number ofXAphase half-metals have beenpredicted based on theoretical simulations [12], we found thatmost of them do not remain in the XA phase after relaxation,or they have lower energies in the L21 or the tetragonal “L21”phase. However, in order to better understand the XA phase, wehave listed in Tables II and III all of the half-metals and near-half-metals found in our survey of 405 XA systems for whichthe calculated formation energy was negative. Our definitionof a “near-half-metal” is given in Sec. IV B.

Both tables are divided into three sections according towhether the gap leading to half-metallicity is located after 9,12, or 14 states. The tables give NV , the number of valenceelectrons per formula unit, a, the lattice constant, the totalmagnetic moment per formula unit as well as the individualmoments on each atom evaluated within atom centered spheresof 1.45-A radius.

For almost all of the half-metals and near-half-metals thatfollow the Slater-Pauling rule (gap after 12 states), the spinmoments on the X1 atoms are antiparallel to those on X2

atoms. If one uses the local moments given in Table II to

094410-12


TABLE II. DFT-calculated properties of 51 half-metals in the inverse Heusler phase that have negative formation energy. Successive columnspresent number of valence electrons per formula unit NV , calculated lattice constant a, total spin moment per f.u. Mtot , local moments for atomson the X1, X2, Y , and Z sites: m(X1), m(X2), m(Y ), and m(Z), formation energy �Ef , distance from the convex hull �EHD, formation energyin L21 phase �Ef (L21), band gap Eg , gap type, experimental reports of compounds with composition X2YZ, and experimental reports ofcorresponding Y2XZ full Heusler compounds, if any.

a Mtot m(X1) m(X2) m(Y ) m(Z) �EHD �Ef (L21) Eg Experimental Y2XZ

X2YZ NV (A) (μB ) �Ef (eV/atom) (eV) reports reports

Sc2VIn 14 6.938 − 4 − 0.072 − 0.509 − 2.675 0.041 − 0.013 0.276 − 0.100 0.504Sc2VSi 15 6.53 − 3 0.058 − 0.452 − 2.228 0.073 − 0.125 0.388 − 0.196 0.499Sc2VGe 15 6.61 − 3 0.084 − 0.424 − 2.274 0.070 − 0.118 0.446 − 0.203 0.480Sc2VSn 15 6.87 − 3 0.144 − 0.327 − 2.388 0.058 − 0.157 0.323 − 0.210 0.518Sc2CrAl 15 6.67 − 3 0.295 − 0.064 − 3.133 0.082 − 0.039 0.225 − 0.096 0.684Sc2CrGa 15 6.62 − 3 0.284 − 0.050 − 3.121 0.067 − 0.065 0.283 − 0.158 0.684Sc2CrIn 15 6.86 − 3 0.337 0.056 − 3.302 0.049 − 0.003 0.286 − 0.069 0.646Sc2CrSi 16 6.43 − 2 0.380 0.020 − 2.454 0.079 − 0.120 0.393 − 0.161 0.632Sc2CrGe 16 6.52 − 2 0.429 0.112 − 2.613 0.065 − 0.110 0.453 − 0.184 0.646Sc2CrSn 16 6.79 − 2 0.505 0.234 − 2.878 0.051 − 0.136 0.344 − 0.180 0.631Sc2MnSi 17 6.37 − 1 0.465 0.443 − 2.218 0.062 − 0.200 0.350 − 0.285 0.518Sc2MnGe 17 6.461 − 1 0.536 0.554 − 2.455 0.041 − 0.188 0.410 − 0.314 0.428Sc2MnSn 17 6.75 − 1 0.628 0.654 − 2.833 0.031 − 0.202 0.312 − 0.287 0.266Ti2VSi 17 6.173 − 1 0.575 − 0.028 − 1.476 0.007 − 0.316 0.217 − 0.379 0.219Ti2VGe 17 6.262 − 1 0.688 0.094 − 1.706 0.008 − 0.246 0.240 − 0.316 0.285Ti2VSn 17 6.523 − 1 0.879 0.257 − 2.052 0.004 − 0.166 0.139 − 0.189 0.423Sc2CoSi 19 6.28 1 0.426 0.173 0.220 0.053 − 0.268 0.406 − 0.506 0.67Sc2CoGe 19 6.367 1 0.432 0.221 0.165 0.029 − 0.248 0.425 − 0.521 0.591Sc2CoSn 19 6.627 1 0.421 0.206 0.182 0.010 − 0.239 0.345 − 0.421 0.594Ti2MnSi 19 6.024 1 1.055 0.652 − 0.920 0.043 − 0.394 0.231 − 0.409 0.461Ti2MnGe 19 6.123 1 1.204 0.844 − 1.314 0.024 − 0.295 0.265 − 0.346 0.517Ti2MnSn 19 6.39 1 1.41 1.106 − 1.916 0.017 − 0.188 0.188 − 0.178 0.593Ti2FeAl 19 6.12 1 1.013 0.709 − 0.964 0.031 − 0.256 0.187 − 0.279 0.536Ti2FeGa 19 6.122 1 1.005 0.757 − 0.966 0.009 − 0.265 0.212 − 0.344 0.57Ti2FeIn 19 6.37 1 1.208 1.064 − 1.673 0.004 − 0.045 0.274 − 0.072 0.532Ti2FeSi 20 5.99 2 1.257 0.621 − 0.101 0.041 − 0.395 0.295 − 0.447 0.628Ti2FeGe 20 6.07 2 1.282 0.677 − 0.186 0.011 − 0.297 0.307 − 0.394 0.622Ti2FeSn 20 6.339 2 1.356 0.801 − 0.509 0.001 − 0.185 0.256 − 0.227 0.617Ti2CoAl 20 6.13 2 1.252 0.705 − 0.204 0.020 − 0.287 0.185 − 0.293 0.684Ti2CoGa 20 6.11 2 1.230 0.773 − 0.211 − 0.005 − 0.295 0.178 − 0.363 0.689Ti2CoIn 20 6.351 2 1.245 0.837 − 0.381 − 0.009 − 0.089 0.195 − 0.086 0.613V2CrGe 20 5.94 2 1.167 − 0.415 1.158 0.003 − 0.118 0.151 − 0.120 0.086V2MnAl 20 5.922 2 1.354 − 0.309 0.870 0.016 − 0.148 0.097 − 0.072 0.142 [74]V2MnGa 20 5.924 2 1.229 − 0.416 1.138 − 0.007 − 0.118 0.123 − 0.070 0.124 [75]Ti2CoSi 21 6.02 3 1.528 0.789 0.394 0.059 − 0.387 0.307 − 0.437 0.798Ti2CoGe 21 6.10 3 1.518 0.812 0.396 0.018 − 0.299 0.303 − 0.384 0.779Ti2CoSn 21 6.35 3 1.495 0.802 0.373 − 0.004 − 0.217 0.220 − 0.197 0.727Ti2NiAl 21 6.19 3 1.524 0.982 0.149 0.028 − 0.285 0.200 − 0.277 0.46Ti2NiGa 21 6.17 3 1.491 1.056 0.158 − 0.003 − 0.289 0.193 − 0.364 0.522Ti2NiIn 21 6.4 3 1.452 1.058 0.135 − 0.013 − 0.127 0.216 − 0.096 0.418

Cr2MnAl 22 5.831 −2 − 1.835 1.536 − 1.678 − 0.021 − 0.086 0.048 0.002 0.204Cr2NiAl 25 5.7 1 − 1.239 1.828 0.452 − 0.057 − 0.071 0.236 0.148 0.142 [76]Cr2CoP 26 5.619 2 − 0.794 1.912 0.854 − 0.012 − 0.22 0.217 − 0.159 0.48Mn2CoAl 26 5.735 2 − 1.617 2.701 0.958 − 0.061 − 0.27 0.036 − 0.141 0.38 XA [28,40] [77]Mn2FeP 27 5.558 3 − 0.452 2.632 0.733 0.024 − 0.352 0.096 − 0.292 0.252Mn2FeAs 27 5.72 3 − 0.732 2.772 0.885 0.022 − 0.071 0.086 0.014 0.337Mn2CoSi 27 5.621 3 − 0.548 2.670 0.849 − 0.012 − 0.365 0.018 − 0.177 0.513 [78–80]Mn2CoGe 27 5.75 3 − 0.804 2.863 0.903 0.005 − 0.153 0.03 − 0.011 0.323 XA [40] [81]Mn2CoP 28 5.581 4 0.091 2.793 1.017 0.032 − 0.333 0.216 − 0.232 0.508Mn2CoAs 28 5.738 4 − 0.007 2.916 1.013 0.025 − 0.085 0.08 0.081 0.295Mn2CoSb 28 5.985 4 − 0.041 3.093 0.938 − 0.003 0.042 0.133 0.193 0.499 XA [40,70] [76,82]

Mn2CuSi 29 5.762 1 − 1.876 2.838 0.003 0.018 − 0.101 0.177 − 0.091 0.318

094410-13


TABLE III. DFT-calculated properties of 50 near-half-metallic X2YZ inverse Heusler compounds with negative formation energy.Successive columns present number of valence electrons per formula unit NV , calculated lattice constant (acalc), total spin moment (Mtot)per f.u., local moments for atoms on the X1, X2, Y , and Z sites: m(X1), m(X2), m(Y ), and m(Z), formation energy (�Ef ), distance from theconvex hull �EHD, formation energy in L21 phase �Ef (L21), experimental reports of compounds with composition X2YZ, and experimentalreports of corresponding Y2XZ full Heusler compounds.

acalc Mtot�Ef �EHD �Ef (L21) Experimental Rep.

X2YZ NV (A) (μB ) m(X1) m(X2) m(Y ) m(Z) (eV/atom) records Y2XZ

Sc2VAl 14 6.756 −3.8779 −0.077 −0.508 −2.588 0.040 −0.033 0.231 −0.134Sc2VGa 14 6.708 −3.9989 −0.114 −0.579 −2.613 0.050 −0.064 0.285 −0.184Sc2MnAl 16 6.59 −1.9983 0.429 0.231 −2.903 0.074 −0.122 0.174 −0.160Sc2MnGa 16 6.54 −1.9999 0.425 0.285 −2.919 0.052 −0.159 0.229 −0.243Sc2MnIn 16 6.80 −1.9969 0.489 0.398 −3.203 0.038 −0.085 0.221 −0.138Sc2VSb 16 6.78 −2.0011 0.319 −0.188 −1.949 0.049 −0.150 0.439 −0.173Ti2VAl 16 6.32 −1.9633 0.380 −0.579 −1.541 0.016 −0.120 0.153 −0.179Ti2VGa 16 6.27 −1.9529 0.314 −0.567 −1.499 0.022 −0.145 0.189 −0.219Sc2CrP 17 6.303 −1.0043 0.491 0.349 −2.051 0.063 −0.166 0.699 −0.296Sc2CrSb 17 6.74 −1.0001 0.660 0.526 −2.598 0.046 −0.097 0.492 −0.178Sc2FeAl 17 6.47 −0.9071 0.248 0.216 −1.539 0.029 −0.1323 0.216 −0.244Sc2FeGa 17 6.43 −0.9488 0.264 0.277 −1.656 0.015 −0.1735 0.267 −0.345Sc2FeIn 17 6.69 −0.9714 0.344 0.412 −2.042 0.013 −0.069 0.290 −0.189Ti2CrAl 17 6.297 −0.8021 1.032 0.775 −2.723 0.019 −0.1556 0.141 −0.197Ti2CrGa 17 6.266 −0.8659 1.008 0.785 −2.747 0.011 −0.1781 0.166 −0.245Ti2CrIn 17 6.51 −0.8538 1.141 0.939 −3.078 0.005 −0.002 0.149 0.022Sc2MnSb 18 6.685 0.0047 −0.656 −0.712 1.984 0.018 −0.1449 0.444 −0.296Ti2VP 18 6.06 0.0033 −0.941 −0.365 1.371 −0.013 −0.4397 0.442 −0.495Sc2NiAl 19 6.502 0.9649 0.413 0.217 0.120 0.031 −0.2413 0.249 −0.425Sc2NiGa 19 6.453 0.9738 0.423 0.272 0.081 0.015 −0.2817 0.278 −0.544Sc2NiIn 19 6.69 0.9839 0.414 0.271 0.078 0.006 −0.1993 0.309 −0.375Ti2CrAs 19 6.11 0.8077 1.015 0.535 −0.941 0.023 −0.2035 0.444 −0.255Ti2CrSb 19 6.38 0.965 1.301 0.775 −1.441 0.019 −0.111 0.279 −0.084Sc2CoSb 20 6.64 1.9999 0.656 0.299 0.664 0.055 −0.138 0.528 −0.394Sc2NiSi 20 6.38 1.8015 0.689 0.459 0.220 0.093 −0.203 0.527 −0.591Sc2NiGe 20 6.46 1.8123 0.691 0.517 0.184 0.050 −0.200 0.540 −0.621Sc2NiSn 20 6.72 1.8438 0.690 0.523 0.164 0.019 −0.228 0.430 −0.516Ti2MnSb 20 6.323 1.9891 1.318 0.564 −0.231 0.009 −0.1596 0.277 −0.181Ti2FeAs 21 6.07 2.9395 1.424 0.469 0.780 0.036 −0.2374 0.475 −0.391V2FeGa 21 5.91 2.8487 1.803 −0.231 1.200 −0.001 −0.146 0.091 −0.047 [76]Ti2NiSn 22 6.427 3.8602 1.604 1.307 0.451 0.001 −0.189 0.294 −0.236

Ti2FeSb 21 6.31 2.9885 1.449 0.441 0.802 0.009 −0.1583 0.319 −0.209V2FeAl 21 5.914 2.911 1.862 −0.197 1.132 0.026 −0.191 0.088 −0.023 [83]

V2MnGe 21 5.92 −2.8136 −1.627 0.516 −1.610 0.005 −0.193 0.115 −0.110V2MnSn 21 6.20 −2.9604 −1.690 0.944 −2.148 0.014 −0.002 0.188 0.179V2FeGe 22 5.891 −1.9779 −1.415 0.434 −0.944 −0.004 −0.165 0.146 −0.129Cr2MnGa 22 5.832 −1.993 −1.880 1.614 −1.716 −0.003 −0.004 0.049 0.038Cr2MnGe 23 5.8 −0.9919 −1.478 1.610 − 1.147 0.010 −0.007 0.122 −0.008V2CoSi 23 5.78 −0.997 −0.754 0.182 −0.374 −0.029 −0.291 0.199 −0.193Cr2CoAl 24 5.79 0.0095 1.892 −1.698 −0.307 0.086 −0.08 0.227 0.126 [84]Cr2CoSi 25 5.657 1.0086 −0.949 1.440 0.547 0.053 −0.2187 0.139 −0.022Cr2CoGe 25 5.792 1.0225 −1.593 2.034 0.619 0.043 −0.0171 0.109 0.155Mn2FeAl 25 5.75 1.0009 −1.856 2.714 0.142 −0.025 −0.158 0.008 −0.086 [75]Mn2FeGa 25 5.79 1.0386 −2.185 2.888 0.315 −0.004 −0.075 0.018 −0.038Cr2NiSi 26 5.705 1.8915 −0.857 2.130 0.573 −0.023 −0.195 0.201 −0.035Cr2NiGe 26 5.82 1.9414 −1.132 2.434 0.573 −0.008 −0.012 0.176 0.124Mn2CoGa 26 5.76 2.0045 −1.820 2.879 0.955 −0.030 −0.167 0 −0.087 XA [40] [85]Mn2FeSi 26 5.6006 2.0077 −0.798 2.407 0.371 −0.015 −0.362 0 −0.259 [76]Mn2FeGe 26 5.72 2.0135 −1.243 2.667 0.556 −0.004 −0.14 0 −0.035

Mn2CuP 30 5.716 1.9946 −1.242 3.042 0.057 0.083 −0.048 0.399 −0.091

094410-14


FIG. 13. Calculated total magnetic moment per unit cell as a function of the total valence electron number per unit cell for 254 XA compoundswith negative formation energies. The orange dashed line represents |Mtot| = NV − 18, the blue dashed line represents the Slater-Pauling rule|Mtot| = NV − 24, and the light green dashed line represents |Mtot| = NV − 28. The compounds from Table II are listed in the boxes. All ofthese compounds follow the rules precisely. We used different colors to label different X-based inverse Heusler compounds. We used diamond,cubic, circle, and triangle symbols to label ferromagnets/ferrimagnets, half-metals, metals, and semiconductors, respectively.

estimate the number of majority and minority s-d electrons onthe transition-metal sites one finds that for the Slater-Paulinghalf-metals with minority gap, there are approximately fourminority electrons on the X1 and Y sites and approximatelytwo on the X2 sites. This implies that the transition-metalnearest neighbors always differ in number of electrons byapproximately 2. The implied contrast in the positions of the d

states of these neighbors leads to the hybridization gap in theminority channel. Since the average of 4 and 2 is 3, the gapshould fall after 3 states per atom [73].

The tables also give the formation energy �Ef and the hulldistance calculated within the OQMD, �EHD (see Sec. IV A 4)as well as the formation energy of the L21 phase, Ef (L21). Thefinal three columns show the calculated energy gap, referencesto experimental reports of the XA phase and references toexperimental reports of the L21 phase for Y2XZ. The presenceof these phases indicates that it may be difficult to distinguish amixture of X2YZ in the XA phase and Y2XZ in the L21 phase.Since they have the same XRD reflection peaks, it is difficultto distinguish the composition from the XRD patterns, whichmay lead to a wrong conclusion about the composition of thesample.

3. Moment vs electron count

Figure 13 shows the total spin magnetic moment performula unit (Mtot) plotted versus the total valence electronnumber per formula unit (NV ) for 254 XA compounds withnegative formation energy. In this plot, the half-metallic phasesfall along one of the lines, Mtot = |NV − 18|, Mtot = |NV −24|, or Mtot = |NV − 28| depending on whether the systemis placing the Fermi energy in a gap after 9, 12, or 14 states,respectively. The half-metals are listed in Table II along withtheir calculated properties. Table III is a similar list of thenear-half-metals.

The half-metals and near-half-metals found along theMtot = |NV − 18| line with a gap after 9 states per formula unit

in one of the spin channels primarily have X = Sc or Ti, whilethose found along the Mtot = |NV − 24| lines generally haveX = Cr or Mn. Systems with X = V also fall mainly alongthe Mtot = |NV − 18| line, but there is one near-half-metal(V2CoSi) that uses the gap after 12 states. We find onlyone half-metal (Mn2CuSi) and one near-half-metal (Mn2CuP)along the Mtot = |NV − 28| line. The systems that appear tobe on that line at NV = 30 and 31 have pseudogaps rather thangaps near the Fermi energy.

Some systems appear on or near two lines. Systems withNV = 21 can in principle have gaps after both 9 states and 12states. Ti2CoSi is a half-metal with the Fermi energy in theminority gap after 9 states, but the Fermi energy also falls in apseudogap at 12 states in majority. The near-half-metals Ti2Fe(P,As,Sb) all have gaps after 9 states in minority and after 12states in majority, however, the moment is not large enoughto place the Fermi energy in the gaps. In each case it fallsjust above the minority gap and just below the majority gapby an equal number of states. Similarly, for NV = 26 one canhave systems on both the Mtot = |NV − 24| and the Mtot =|NV − 28| lines. Thus, Mn2CoAl has a minority gap after 12states and a pseudogap at 14 majority states.

Although we predicted that Mn2CoSb would have a positiveformation energy (0.042 eV), we found references indicatingthat it can be fabricated in the XA phase [40,70] usingmelt spinning. For this reason, Mn2CoSb was included inFig. 13 and Table II. As discussed in Sec. III A, we speculatethat the experimental samples may be L21B phase ratherthan XA.

4. Hull distances

Using our calculated XA formation energies, and the for-mation energies of all the other phases in the OQMD, we calcu-lated the hull distance �EHD for the negative formation energyhalf-metallic inverse Heusler compounds listed in Table IIand the near-half-metals listed in Table III. Our calculations

094410-15


predict all the Sc-, Ti-, and V-based half-metals lie significantlyabove the convex hull and have a �EHD that exceeds ourempirical criterion of �EHD � 0.052 eV/atom for having areasonable probability of synthesizability. We predict fourhalf-metals (Cr2MnAl, Mn2CoAl, Mn2CoSi, Mn2CoGe) to lieclose enough to the convex hull to meet our criterion. Twoof these compounds Mn2CoAl [28,40] and Mn2CoGe [40]have been experimentally observed by x-ray diffraction (XRD)measurements to exist in the XA phase. We also found sixnear-half-metals (Cr2MnGa, Mn2FeAl, Mn2FeGa, Mn2FeSi,Mn2FeGe, Mn2CoGa) that meet our criterion for stability,three of these (Mn2FeGa, Mn2FeSi, Mn2FeGe) have a cal-culated hull distance of zero, meaning that no phase orcombination of phases in the OQMD had lower energy.

B. Electronic structure of each IH compound

In this section we will briefly discuss each of the 405 XA

compounds discussing gaps in their DOS, and whether theyare half-metals or near-half-metals. The designation of half-metal is binary. If the Fermi energy lies in a gap for one ofthe spin channels, but not for the other, then it is a half-metal.There may be other important quantitative questions such as thedistances between the Fermi energy and the band edges, but thecalculated band structure either corresponds to a half-metal orit does not. It should be noted that the determination of whethera given inverse Heusler has a gap is not always trivial if thegap is small because the Brillouin zone is sampled at a finitenumber of points. We used both the DOS calculated using thelinear tetrahedron method and an examination of the bandsalong symmetry directions to decide difficult cases.

In contrast to the binary choice between half-metal or not,the designation of a “near-half-metal” is more qualitative.There are a couple of ways a calculated DOS could be “almost”half-metallic: (a) the Fermi energy could fall at an energy forwhich the DOS for one of the spin channels is very small,i.e., a pseudogap, (b) the DOS could have a gap in one of thespin channels with the Fermi energy lying slightly outside thegap. If we choose criterion (a), we must choose a threshold for“very small,” and if we choose (b), we must choose a thresholdfor “slightly.” Criterion (a) is difficult to implement becausethere are an infinite number of ways the DOS can vary near apseudogap, e.g., a pseudogap may be wide and only fairly smallor it might be narrow but extremely small or some combination.Another issue with criterion (a) is that a low density of statesis often associated with a highly dispersive band that leads to alarge conductivity. For these reasons, we choose criterion (b),i.e., we require that there be an actual gap and that the Fermienergy fall “near” this gap. The (arbitrary) criterion we choosefor “near” is 0.2 electron states per formula unit. Other criteriacould be used, e.g., the distance from the Fermi energy to thenearest band edge. Our choice of using the number of states hasthe advantage of being easily related to the magnetic moment.It has the disadvantage that occasionally a system with a highlydispersive band above or below the Fermi energy with a verysmall density of states can lead to a situation in which we labela system near-half-metallic when the Fermi energy is as muchas 0.3 eV into the band.

For each of the systems, we note the gaps in the DOS,typically after 9, 12, or 14 states per formula unit per spin

channel and whether the Fermi energy falls in one or more ofthe gaps or close enough to the gap (0.2 states) to be designateda near-half-metal. Simple state and electron counting dictatesthat to make a half-metal by taking advantage of a gap afterNs states when there are N electrons per formula unit, themagnetic moment per formula unit must be N − 2Ns .

13-electron systems. Our data set contained 3 systems with13 valence electrons, Sc2Ti(Al,Ga,In). All three were magneticmetals with small gaps after 9 states per spin channel in bothchannels (except for Sc2TiIn minority). However, the momentswere not large enough (a total moment of 5μB per formula unitwould have been needed) to move the majority Fermi energyinto the gap.

14-electron systems. Our data set contained 6 systems with14 valence electrons, Sc2Ti(Si,Ge,Sn) and Sc2V(Al,Ga,In).These systems require a moment of 4μB to place the Fermienergy in a majority gap after 9 electrons per spin channel.

Sc2Ti(Si,Ge,Sn): Sc2TiSi and Sc2TiGe are predicted to benonmagnetic metals and to have small gaps after 9 states inboth spin channels. Sc2TiSn is predicted to be magnetic andto have a majority gap after 9 states, but its moment closer to3μB than 4μB .

Sc2V(Al,Ga,In): These all have relatively large gaps after9 states in the majority channel. The position of the Fermienergy rises relative to the gap as the atomic size of the Z

atom increases the lattice constant, facilitating the formationof large moments on V and even small moments on Sc. ForSc2VAl and Sc2VGa, the moment is not quite large enoughto place the Fermi energy in the gap making them near-half-metals (Table III). However, the lattice constant of Sc2VIn islarge enough to support a moment large enough to move theFermi energy into the gap so it is a half-metal (Table II).

15-electron systems. Our data set contained 9 systems with15 valence electrons, Sc2Ti(P,As,Sb), Sc2V(Si,Ge,Sn), andSc2Cr(Al,Ga,In). A moment of 3μB is needed to place theFermi energy in a majority gap after 9 states.

Sc2Ti(P,As,Sb): These systems do not have gaps after 9states per atom in either channel because of a reordering ofthe states at �. A singly degenerate state, usually band 10 at�, interchanges with a triply degenerate T2 state so that bands8, 9, and 10 are associated with the triply degenerate T2 state,precluding a gap.

Sc2V(Si,Ge,Sn): These systems have majority gaps after 9states. The Fermi energy is in the gap and they are classifiedas half-metals as described in Table II.

Sc2Cr(Al,Ga,In): These systems all have large gaps after 9states in majority and small gaps or pseudogaps near 9 statesin minority. The Fermi energy falls in the majority gap forall three so they are classified as half-metals and described inTable II.

16-electron systems. Our data set contained 12 systemswith 16 valence electrons per formula unit, Sc2V(P,As,Sb),Sc2Cr(Si,Ge,Sn), Sc2Mn(Al,Ga,In), and Ti2V(Al,Ga,In). For16 valence electrons per formula unit and a gap after 9 statesin the majority-spin channel, a moment of 2μB is needed toplace the Fermi energy in the gap to make a half-metal.

Sc2V(P,As,Sb): Sc2VP and Sc2VAs have a region with anextremely low but nonzero DOS at the Fermi energy. Thesepseudogaps can be be traced to a band reordering similar tothat mentioned previously in which a singly degenerate state

094410-16


at � drops below a triply degenerate state so that the triplydegenerate state connects to bands 8, 9, and 10, thus precludinga gap. In the case of Sc2VSb, the singly degenerate state is afew hundredths of an eV above the triply degenerate state so itis a near-half-metal with a tiny gap.

Sc2Cr(Si,Ge,Sn): These systems have large majority gapsafter 9 states. They generated half-metals as described inTable II.

Sc2Mn(Al,Ga,In): The three systems with X = Sc andY = Mn have large majority and tiny minority gaps after 9states. Their moments are very slightly less than 2μB so theygenerated near-half-metals as described in Table III by placingthe Fermi energy at the valence edge of the majority gap.

Ti2V(Al,Ga,In): These systems have tiny majority andminority gaps after 9 states per spin channel. All take advantageof the majority gap to generate near-half-metals as shown inTable III. Ti2VIn is not included in the table because it has apositive formation energy.

17-electron systems. Our data set contained 15 systemsthat had 17 electrons per formula unit Sc2Cr(P,As,Sb),Sc2Mn(Si,Ge,Sn), Sc2Fe(Al,Ga,In), Ti2V(Si,Ge,Sn), andTi2Cr(Al,Ga,In). Systems with 17 electrons per formula unitcan generate half-metals by taking advantage of a gap after 9states in majority if their magnetic moment per formula unit is1μB .

Sc2Cr(P,As,Sb): Sc2CrP and Sc2CrAs both have majoritypseudogaps at the Fermi energy. The gap after 9 electronsper formula unit is converted into a pseudogap by a bandreordering at � similar to that occurring for Sc2V(P,As) andSc2Ti(P,As,Sb). Sc2CrSb has a “normal” band ordering at �

and is a near-half-metal similar to Sc2VSb, but with a muchlarger gap.

Sc2Mn(Si,Ge,Sn): These systems have majority gaps after9 states and are all half-metals as described in Table II. Inaddition to the majority gap, Sc2MnSi has a gap after 9 statesin minority. A pseudogap near 9 states in minority for Sc2MnSican also be identified.

Sc2Fe(Al,Ga,In): These systems have gaps after 9 states inboth majority and minority channels. All three take advantageof the majority gap to generate near-half-metals as describedin Table III. Sc2FeAl also has a minority pseudogap near 12states and a majority pseudogap near 14 states.

Ti2V(Si,Ge,Sn): These all have gaps after 9 states in bothmajority and minority with the majority gap being larger. Allthree compounds utilize the majority gap to form half-metalsas described in Table II.

Ti2Cr(Al,Ga,In): These have gaps after 9 states in bothspin channels. They are all near-half-metals (Table III), havingmoments just large enough to satisfy our criterion. The largemoment on the Cr atom opens a large gap in the majoritychannel (hybridization between Cr and Ti-d and with the Cr-donsite energy well below that of the Ti atoms). The gap inminority is smaller and is dependent on the difference betweenthe two Ti potentials and avoided band crossings.

18-electron systems. Our data set contained 18 systems with18 valence electrons. These were described in Sec. IV A 1.

19-electron systems. Our data set contained 21 systemswith 19 valence electrons per formula unit Sc2Fe(P,As,Sb),Sc2Co(Si,Ge,Sn), Sc2Ni(Al,Ga,In), Ti2Cr(P,As,Sb),Ti2Mn(Si,Ge,Sn), Ti2Fe(Al,Ga,In), and V2Cr(Al,Ga,In).

Systems with 19 valence electrons need a moment of 1μB toplace the Fermi energy in a gap after 9 states in the minoritychannel.

Sc2Fe(P,As,Sb): Sc2FeP is nonmagnetic with gaps after 9states in both spin channels. Sc2FeAs is also nonmagnetic,but the gaps after 9 states are converted to pseudogaps by theinversion of singly and triply degenerate states at � similar toSc2CrP and Sc2CrAs. The larger lattice constant of Sc2FeSballows it to develop a moment. There are sizable gaps after9 states in both the minority and majority channels, but themoment is not quite large enough (0.79μB) to meet ourcriterion for calling it a near-half-metal.

Sc2Co(Si,Ge,Sn): All three of these are half-metals withsizable gaps after 9 states in both spin channels (Table III).The Fermi energy falls near the top of the gap in all threecases, but especially for Ge.

Sc2Ni(Al,Ga,In): All three have sizable gaps after 9 statesin both spin channels and moments as given in Table III thatqualify them as near-half-metals.

Ti2Cr(P,As,Sb): All three compounds have gaps in theminority channel after 9 states, but only Ti2CrAs and Ti2CrSbhave sufficiently large moments to be classified as near-half-metals. (Ti2CrAs only barely made the cutoff.) Ti2CrSb alsohas a tiny gap after 9 states in majority.

Ti2Mn(Si,Ge,Sn): All three of these compounds have siz-able gaps after 9 states in minority and much smaller gaps inmajority. They also have pseudogaps near 12 states in minority.For all three, the Fermi energy falls in the gap so they are areincluded in Table III.

Ti2Fe(Al,Ga,In): All three of these compounds have gapsafter 9 states in both channels and pseudogaps in both channelsafter 12 states. The gaps after 9 states are much smaller inmajority than in minority. The Fermi energy for each of thethree falls in the minority gap creating three half-metals asdescribed in Table II.

V2Cr(Al,Ga,In): All three of these systems have gaps after 9states in both spin channels. In all cases, however, the magneticmoment is not large enough to meet our criterion for a near-half-metal. The formation energy is small and negative forV2CrAl and V2CrGa. It is positive for V2CrIn.

20-electron systems. Our data set contained 24 compoundswith 20 electrons per formula unit Sc2Co(P, As, Sb), Sc2Ni(Si,Ge, Sn), Sc2Cu(Al, Ga, In), Ti2Mn(P, As, Sb), Ti2Fe(Si, Ge,Sn), Ti2Co(Al,Ga,As), V2Cr(Si, Ge, Sn), and V2Mn(Al, Ga,In).

Sc2Co(P,As,Sb): Sc2CoP and Sc2CoAs both have gaps after9 states per formula unit in both spin channels, but in bothcases the net magnetic moment is much too small to pull theFermi energy into one of the gaps. Sc2CoSb, on the otherhand, due to its larger lattice constant, supports a momentsufficiently large (1.999μB ) that the Fermi energy falls at thetop edge of the minority gap. We classify it as a near-half-metal(Table III).

Sc2Ni(Si,Ge,Sn): The compounds Sc2Ni(Si,Ge,Sn) all havegaps after 9 electrons in both spin channels. However, the totalmoment (1.80μB , 1.81μB , and 1.84μB , respectively) does notreach the value (2.00μB) needed to place the Fermi energyin the gap. Even when the lattice is expanded, the magneticmoment of these Sc2Ni(Si,Ge,Sn) compounds hardly increasesbecause the d bands of Ni are filled so that Ni only supports

094410-17


a very small moment in these materials. Nevertheless, thesecompounds meet (barely) our threshold to be called near-half-metals (Table III).

Sc2Cu(Al,Ga,In): The systems Sc2Cu(Al,Ga,In) all havepseudogaps after 9 states in both spin channels.

Ti2Mn(P,As,Sb): All three of these compounds have gapsafter 9 electrons in both spin channels. However, only Ti2MnSb(due to its larger lattice constant) has a moment (1.989μB )close enough to 2μB to be classified a “near-half-metal.”Ti2MnAs (1.7967μB ) barely misses the cutoff to be includedin Table III.

Ti2Fe(Si, Ge, Sn): The compounds Ti2Fe(Si,Ge,Sn) all havegaps in both channels after 9 states and pseudogaps after 12states. The Fermi energy falls within the gap in all three casesso they are predicted to be half-metals as described in Table II.

Ti2Co(Al,Ga,As): The compounds Ti2Co(Al,Ga,In) havelarge minority gaps after 9 states and somewhat smaller gapsafter 9 states in majority. The Fermi energy falls within theminority gap in all three cases so they are are predicted to behalf-metals as described in Table II.

V2Cr(Si,Ge,Sn): V2CrSi has small gaps after 9 states in bothspin channels, but is not magnetic. V2CrGe is magnetic with amoment of 2μB . The Fermi energy falls very close to the top ofthe gap, but V2CrGe is predicted to be a half-metal as describedin Table II. V2CrSn has a gap after 9 states in the majoritychannel which is not useful for generating a half-metal. Thereis no gap in minority.

V2Mn(Al, Ga, In): V2MnAl has a tiny majority gap anda sizable minority gap after 9 states. It also has pseudogapsafter 12 states in majority and 14 states in minority. It takesadvantage of the minority gap after 9 states to become a half-metal. The DOS for V2MnGa is similar. It is also a half-metal.Both compounds are included in Table II. V2MnIn has a gapafter 9 states in the majority, but only a pseudogap after 9states in minority. Its large moment, in excess of 3μB, placesthe Fermi energy well below this pseudogap.

21-electron systems. Our database contains 27 systemswith 21 valence electrons Sc2Ni(P,As,Sb), Sc2Cu(Si,Ge,Sn),Sc2Zn(Al,Ga,In), Ti2Fe(P,As,Sb), Ti2Co(Si,Ge,Sn),Ti2Ni(Al,Ga,In), V2Cr(P,As,Sb), V2Mn(Si,Ge,Sn),V2Fe(Al,Ga,In). Systems with 21 valence electrons areinteresting because they can, in principle, place the Fermienergy in a gap after 9 electrons or a gap after 12 electronsor both. If a majority gap present after 12 states is present aswell as a minority gap after 9 states, a moment of 3μB wouldyield a magnetic semiconductor.

Sc2Ni(P,As,Sb): Sc2NiP has tiny gaps in both spin channelsafter 9 states while Sc2NiAs has pseudogaps after 9 states inboth spin channels. Both have very small moments. Sc2NiSbhas gaps after 9 states in both spin channels, but its moment(2.15μB ) is too small to place the Fermi energy in or near thegap.

Sc2Cu(Si,Ge,Sn): All three of these compounds have gapsafter 9 states in minority and much smaller gaps in majority.However, their moments are too small to place the Fermi energyin the minority gap.

Sc2Zn(Al,Ga,In): These three nonmagnetic compoundsonly have pseudogaps after 9 states in both spin channelsbecause of a highly dispersive band 10 which drops belowband 9 along � to X.

None of the 21-electron systems with X = Sc generate half-metals or near-half-metals because a moment of 3μB would benecessary to take advantage of a gap after 9 states per spin performula unit or to take advantage of a gap after 12 states perspin per formula unit. The compounds are not able to do thisbecause Sc is difficult to polarize and Ni, Cu, and Zn cannotsupport a large moment because their d bands are filled or (inthe case of Ni) nearly filled.

Ti2Fe(P,As,Sb): These three compounds have large gapsafter 9 states in the minority channel and smaller gaps inthe majority channel after 9 states. Ti2FeP and Ti2FeAs havepseudogaps after 12 states in the majority which becomes avery small real gap for Ti2FeSb. The moments for the threeare 2.72μB , 2.94μB , and 2.99μB , respectively. Ti2FeAs andTi2FeSb meet our criterion for near-half-metallicity. If the mo-ment for Ti2FeSb were very slightly larger, we would predictit to be a magnetic semiconductor with a large minority gapand a tiny majority gap, i.e., a 9/12 magnetic semiconductor.The present prediction is for a magnetic semimetal that has apocket of down-spin electrons along � to K and a �-centeredpocket of up-spin holes.

Ti2Co(Si,Ge,Sn): These three compounds all have gaps inboth spin channels after 9 states, with the minority gap beinglarger. Ti2CoSi also has a majority pseudogap after 12 states.For all three, the Fermi energy falls within the minority gapgiving each a moment of 3μB and making them half-metals(Table II). For Ti2CoSi the current at zero K would be carriedby up-spin holes at � and up-spin electrons at X.

Ti2Ni(Al,Ga,In): These three compounds all have gaps after9 states in both spin channels. The Fermi energy falls wellinside the minority gaps yielding half-metals with moments of3μB in each case (Table II).

V2Cr(P,As,Sb): V2CrP has gaps after 9 states in both spinchannels, but is nonmagnetic. V2CrP has a gap in the minoritychannel after 9 states and a pseudogap after 9 states in themajority channel. Its moment is too small (1.85μB ) for it to beclassified as a near-half-metal. V2CrSb has a pseudogap after9 states in the majority channel and a sizable gap after 12 statesalso in majority channel. Its moment is large enough (2.89μB )that we would classify it as a near-half-metal. However, itsformation energy is positive so it is not included in Table III.

V2Mn(Si,Ge,Sn): V2MnSi has pseudogaps after 9 states inboth spin channels and is nonmagnetic. V2MnGe has gapsafter 9 states in both spin channels and a pseudogap after 12states in majority. Its moment of 2.81μB qualifies it (barely)for inclusion with the near-half-metals in Table III. V2MnSnhas majority gaps after 9 and 12 states and a pseudogap after9 states in minority. Its moment of 2.96μB qualifies it forinclusion in Table III as a Slater-Pauling near-half-metal.

V2Fe(Al,Ga,In): V2FeAl has gaps after 9 and 14 states inminority and after 12 states in majority. Its moment of 2.91μB

qualifies it as a “double near-half-metal” since its Fermi energyis near gaps after 9 states in minority and 12 states in majority.V2FeGa has a gap after 9 states in minority and a pseudogapafter 12 states in majority. Its moment of 2.85μB meets thethreshold for inclusion in Table III as a near-half-metal. V2FeInhas gaps in both spin channels after 9 states and a pseudogapafter 12 states in majority. Its moment is large enough forinclusion for inclusion in Table III, but its formation energyis positive.

094410-18


22-electron systems. Our database contains 27 systemswith 22 electrons Sc2Cu(P,As,Sb), Sc2Zn(Si,Ge,Sn),Ti2Co(P,As,Sb), Ti2Ni(Si,Ge,Sn), Ti2Cu(Al,Ga,In),V2Mn(P,As,Sb), V2Fe(Si,Ge,Sn), V2Co(Al,Ga,As), andCr2Mn(Al,Ga,In). These systems would need a moment of4μB to place the Fermi energy into a gap after 9 states inthe minority channel or a moment of 2μB to place the Fermienergy into a gap after 12 states in the majority channel.

Sc2Cu(P,As,Sb): The systems with X = Sc and Y = Cugenerate only small moments (0.40, 0.62, and 0.79)μB for Z =P, As, and Sb, respectively. They have gaps in both channelsafter 9 states and no gaps after 12 states.

Sc2Zn(Si,Ge,Sn): The systems with X = Sc and Y = Znalso generate moments that are too small (0.55μB , 0.92μB ,and 0.93μB for Si, Ge, and Sn respectively). These systemshave pseudogaps after 9 states in both spin channels, but nogaps after 12 states.

Ti2Co(P,As,Sb): These three systems have gaps after 9 statesin both spin channels, but their moments (2.20, 2.38, and2.73)μB are much smaller than the 4μB needed to place theFermi energy in the minority gap.

Ti2Ni(Si,Ge,Sn): These three systems have gaps after 9states in both spin channels. Their moments are larger thanthose of the Ti2Co(P,As,Sb) systems (3.47μB , 3.69μB , and3.86μB ) for Si, Ge, and Sn, respectively. Thus, Ti2NiSn meetsour threshold for inclusion in Table III as a near-half-metal.

Ti2Cu(Al,Ga,In): Ti2CuAl has gaps after 9 states in bothspin channels, but its moment is too low (3.72μB ) to meetour criterion for near-half-metal status. Ti2CuGa and Ti2CuInhave gaps after 9 states in majority, but only pseduogaps after9 states in minority.

The 22-electron systems with X = Ti and Y = Co, Ni, orCu tend to have gaps after 9 states per spin channel per formulaunit in both spin channels. They do not have gaps after 12 statesper spin channel, presumably because insufficient contrast canbe generated between the d-onsite energies of the two Ti atoms.In most cases, the moments are too small to place the Fermienergy in the minority gap after 9 states. The moments arelargely on the Ti atoms and the moments tend to increase withthe atomic number of the Y atom, increasing from Co to Ni toCu, and with the size of the Z atom.

V2Mn(P,As,Sb): V2MnP has gaps after 9 states in bothspin channels but a very small moment (0.59μB) so it cannottake advantage of the minority gap to make a half-metal.V2MnAs also has gaps after 9 states in both channels and apseudogap after 12 states in the majority. Its moment of 1.97μB

places the Fermi energy close to the majority pseudogapafter 12 states, but the absence of a real gap precludes itsdesignation as a near-half-metal. V2MnSb has a majoritygap after 9 states, a minority pseudogap after 9 states and amajority gap after 12 states. Its moment of 1.9987μB placesthe Fermi energy just below the majority gap after 12 states.Only its positive formation energy precludes its inclusion inTable III.

V2Fe(Si,Ge,Sn): All three compounds have gaps after 9states in both spin channels. V2FeGe also has a majority gapafter 12 states and V2FeSn has a pseudogap after 12 statesin the majority channel. The moment of V2FeGe (1.98μB )places its Fermi energy quite close to the majority gap so it isa near-half-metal as described in Table III.

V2Co(Al,Ga,As): All three of these compounds have gapsin both channels after 9 states with the minority gap beingmuch larger than the small majority gap. All three also havepseudogaps after 12 states in the majority. The pseudogap forV2CoAl is particularly pronounced. The moment is slightlyless than 2μB for all three compounds, but according to ourdefinition, they are not near-half-metals.

Cr2Mn(Al,Ga,In): Cr2MnAl and Cr2MnGa both have gapsafter 12 states in the majority channel. Cr2MnIn only has apseudogap there. Cr2MnAl is a half-metal as described inTable II and Cr2MnGa is a near-half-metal as described inTable III. All three of these compounds are predicted to haveferrimagnetic states in which the moments of Cr1 and Mn arealigned, and the moment of Cr2 is opposite.

23-electron systems. Our database contains 27 systemswith 23 electrons Sc2Zn(P,As,Sb), Ti2Ni(P,As,Sb),Ti2Cu(Si,Ge,Sn), Ti2Zn(Al,Ga,In), V2Fe(P,As,Sb),V2Co(Si,Ge,Sn), V2Ni(Al,Ga,In), Cr2Mn(Si,Ge,Sn), andCr2Fe(Al,Ga,In). These compounds can take advantage ofa gap after 9 states in minority to make a half-metal, but amoment of 5μB would be needed. Alternatively, they couldtake advantage of a gap after 12 states in majority whichwould only require a moment of 1μB . A few of these systemshave pseudogaps after 14 states. In principle, one could havea 23-electron inverse Heusler magnetic semiconductor with amoment of 5μB that uses gaps after 9 states in minority and14 states in majority; however, such solutions did not appearin our database.

Sc2Zn(P,As,Sb): The three systems Sc2Zn(P,As,Sb) havepseudogaps after 9 states in both channels. They are allnonmagnetic and have no gaps after 12 sates.

Ti2Ni(P,As,Sb): These compounds all have gaps after 9states in both spin channels, but their moments are much toosmall to make half-metals or near-half-metals.

Ti2Cu(Si,Ge,Sn): The three systems Ti2Cu(Si,Ge,Sn) havegaps in both channels after 9 and pseudogaps in majority after14 states. Their moments (3.00, 3.16, and 3.33)μB , however,are much less than the 5μB that would be needed to place theFermi energy in the minority gap.

Ti2Zn(Al,Ga,In): These compounds all have pseudogapsnear 9 states per formula unit in the minority channel butno gaps at 9 or 12 states nor is the Fermi energy near thepseudogap. Ti2ZnAl and Ti2ZnGa also have majority gaps after9 states and Ti2ZnAl also has a pseudogap after 14 states inmajority.

V2Fe(P,As,Sb): These compounds have gaps in both spinchannels after 9 states. V2FeP and V2FeAs have pseudogapsin the majority channel after 12 states. V2FeSb, however, doeshave a majority gap after 12 states and the magnetic moment(0.9996μB ) places the Fermi energy at the lower edge of thisgap. V2FeSb is not included in Table III because it has a positiveformation energy.

V2Co(Si,Ge,Sn): These compounds have gaps after 9 statesin both channels, but only pseudogaps after 12 states except forV2CoSi which has a tiny gap after 12 states. The Fermi energyfalls just below this gap (moment = 0.997μB ). (See Table III.)

V2Ni(Al,Ga,In): These compounds have sizable gaps after 9states in both spin channels, but the moments are small (≈1μB )and there are no gaps after 12 states in the majority channel,hence, no half-metals or near-half-metals.

094410-19


Cr2Mn(Si,Ge,Sn): Cr2MnSi has a gap after 9 states in themajority channel and a pseudogap after 9 states in minority.Cr2MnGe has a pseudogap after 9 states in majority and asizable gap after 12 states in majority. Its moment of 0.99μB

allows it to use this gap to become a near-half-metal (Table III).Cr2MnSn also has a majority gap after 12 states and a momentof approximately 1.00μB so it is a near-half-metal, but itis not included in Table III because its formation energy ispositive.

Cr2Fe(Al,Ga,In): Cr2FeAl has minority gaps after 9 statesand 14 states, and a majority pseudogap near 12 states. Itsmoment of approximately 1.00μB places the Fermi energyinto this pseudogap. Cr2FeGa has a minority pseudogap after9 states and a majority gap after 12 states. Its moment of0.98μB makes it a near-half-metal, but it is not included inTable III because its formation energy is positive. Cr2FeGahas a majority pseudogap near 12 states.

24-electron systems. Our database contained 24 systemswith 24 valence electrons Ti2Cu(P,As,Sb), Ti2Zn(Si,Ge,Sn),V2Co(P,As,Sb), V2Ni(Si,Ge,Sn), V2Cu(Al,Ga,In),Cr2(P,As,Sb), Cr2Fe(Si,Ge,Sn), Cr2Co(Al,Ga,In). Basedon electron count alone, these systems would offer theopportunity for nonmagnetic semiconductors or zero netmoment half-metals arising from Slater-Pauling gaps after12 states per formula unit in one or both spin channels.In practice, we find that only 4 of the 24 systems displaySlater-Pauling gaps (and only in one spin channel). In allfour of these systems the Fermi energy falls at the edge ofthe gap but just outside so that they would be classified asnear-half-metals. Three of these four near-half-metals havepositive formation energy.

The gaps after 9 states, on the other hand, seem to be morerobust. Only Cr2MnSb failed to show a gap or a pseudogapafter 9 states in at least one of the spin channels. Most systemsshowed gaps after 9 states in both spin channels. However,none of the systems were able to generate a moment of 6μB

that would have been necessary to use one of the gaps after 9states to generate a half-metal.

Ti2Cu(P,As,Sb): These systems showed gaps in both chan-nels after 9 states, but their moments were much too small toplace the Fermi energy near the minority gap.

Ti2Zn(Si,Ge,Sn): These systems showed pseudogaps inboth channels near 9 states per formula unit. The gaps wereconverted to pseudogaps by a highly dispersive band 10 along� to X dropping down below the maximum of bands 7–9 at �.

The 9 systems in our database with X = V all showedgaps after 9 states in both spin channels. In all cases, themoments were quite small so that the Fermi energy could notfall into these gaps. Six of the 9 systems V2Co(P,As,Sb) andV2Ni(Si,Ge,Sn) have pseudogaps around 12 states per atom inat least one spin channel.

V2Co(P,As,Sb): These systems show gaps after 9 states andpseudogaps after 12 states in both spin channels. The momentsare too small (V2CoSb) or zero precluding half-metals.

V2Ni(Si,Ge,Sn): V2NiSi has gaps in both spin channelsafter 9 states and pseudogaps after both spin channels after12 states. V2NiGe has gaps in both spin channels after 9 statesand a marginal pseudogap after 12 states in majority. V2NiSnhas gaps after 9 states in both spin channels and pseudogapsafter 12 states in minority and after 14 states in majority. The

lack of real gaps after 12 states and their very small momentspreclude half-metals.

V2Cu(Al,Ga,In): These compounds have gaps after 9 statesin both spin channels and minority gaps after 14 states. V2CuInalso has a majority gap after 14 states while V2CuAl andV2CuGa have pseudogaps in majority near 14 states. Theirsmall moments and the absence of gaps after 12 states precludehalf-metals.

Cr2Mn(P,As,Sb): In the minority channel, Cr2MnP has agap after 9 states and a pseudogap near 12 states. It alsohas a pseudogap near 9 states in majority. It has a smallmoment placing the Fermi energy in the minority pseudogapnear 12 states. Cr2MnAs has minority gaps after 9 and 12states. The very small moment (0.005μB ) places the Fermienergy just below the gap at 12 states. Thus, Cr2MnAs is anear-half-metal, but it is not included in Table III becauseits calculated formation energy is positive. Cr2MnSb has aminority gap after 12 states. Its tiny moment (0.0002μB ) makesit a near-half-metal, but it is not included in Table III becauseits formation energy is positive.

Cr2Fe(Si,Ge,Sn): Cr2FeSi is nonmagnetic and has gapsafter 9 states in both channels. Cr2FeGe has gaps in both spinchannels after 9 states and a minority gap after 12 states. Itssmall moment makes it a near-half-metal, but we do not list itin Table III because of its positive formation energy. Cr2FeSnhas a majority gap after 9 states and a minority gap after 12states. Its small moment places the Fermi energy somewhatbelow this gap yielding a near-half-metal. It is not listed inTable III because of its positive formation energy.

Cr2Co(Al,Ga,In): These systems all have gaps after ninestates in the majority channel and all have gaps (Al,Ga) orpseudogaps (In) in the minority channel after 12 states. Allthree have tiny moments. Cr2CoAl and Cr2CoGa are near-half-metals, but only the former has a negative formation energy.Cr2CoAl is the only Slater-Pauling 24-electron near-half-metalwith negative formation energy in Table III.

25-electron systems. Ti2Zn(P,As,Sb), V2Ni(P,As,Sb),V2Cu(Si,Ge,Sn) V2Zn(Al,Ga,In), Cr2Fe(P,As,Sb),Cr2Co(Si,Ge,Sn), Cr2Ni(Al,Ga,As), Mn2Fe(Al,Ga,As).Many of the 25-electron inverse Heusler systems show gapsafter 9 states per formula in one or both spin channels. Noneof these systems, however, are able to take advantage ofthese gaps to make a half-metal or near-half-metal by movingthe Fermi energy to the gap because that would require arelatively large moment (7μB per formula unit). This largemoment would be difficult to produce in systems whosetransition-metal elements come from the early or late partof the transition-metal series, e.g., X = Ti or V and Y =Ni, Cu, or Zn. The systems that could produce such largemoments, e.g., X = Cr or Mn and Y = Fe or Co seem to prefersmaller net moments closer to 1μB which allows them to takeadvantage of gaps and pseudogaps after 12 states per formulaunit. Several systems show gaps after 14 states, but none ofthese have the correct moment (3μB) to make a half-metal.

Ti2Zn(P,As,Sb): All three compounds have gaps after 9states in both spin channels, but their moments are too smallto move the Fermi energy near these gaps.

V2Ni(P,As,Sb): All three compounds have gaps after 9states in both spin channels, but their moments are too smallto move the Fermi energy near these gaps.

094410-20


V2Cu(Si,Ge,Sn): All three compounds have gaps after 9states in both spin channels. V2CuSn also has gaps after 14electrons in both spin channels. In all three cases, the momentsare too small to move the Fermi energy near the minority gapafter 9 states or the majority gap after 14 states.

V2Zn(Al,Ga,Sb): V2ZnAl has a gap after 14 states inmajority and a pseudogap after 14 states in minority. V2ZnGahas a pseudogap after 14 states in majority and a gap after14 states in minority. V2ZnIn has a gap in both spin channelsafter 14 states. Their small moments preclude half-metals ornear-half-metals.

Cr2Fe(P,As,Sb): Cr2FeP has gaps in both spin channels after9 states and a pseudogap in minority after 12 states. Cr2FeAshas gaps in both spin channels after 9 states and a sizable gapin minority after 12 states. Cr2FeSb has a gap after 9 statesin majority and a pseudogap after 9 states in minority. It alsohas a gap after 12 states in minority. All three have momentsof approximately 1.01μB which places the Fermi energy nearthe gap or pseudogap at 12 states. Cr2FeAs and Cr2FeSb arenear-half-metals, but are not included in Table III because oftheir positive formation energies.

Cr2Co(Si,Ge,Sn): Cr2CoSi has gaps in both spin channelsafter 9 states and a minority gap after 12. Cr2CoGe has amajority gap after 9 states and a minority gap after 12 states.Both of these systems are predicted to be near-half-metals asdescribed in Table III. Cr2CoSn has a majority gap after 9 statesand a pseudogap in minority near 12 states.

Cr2Ni(Al,Ga,In): Cr2NiAl has gaps after 9 and 14 states inmajority and after 12 states in minority. It uses the gap after12 states in minority to generate a half-metal as described inTable II. Cr2NiGa has gaps after 9 states in majority and after14 states in minority. It also has a pseudogap near 12 statesin minority. Cr2NiIn has a gap after 9 states in majority and apseudogap near 12 states in minority.

Mn2Fe(Al,Ga,In): Mn2FeAl and Mn2FeGa both have gapsafter 9 states in majority and after 12 states in minority. Bothplace the Fermi energy slightly below the minority gap to gen-erate near-half-metals as described in Table III. In Mn2FeIn,the minority gap after 12 states becomes a pseudogap. It alsohas minority gaps after 9 states and after 14 states.

26-electron systems. V2Cu(P,As,Sb), V2Zn(Si,GeSn),Cr2Co(P,As,Sb), Cr2Ni(Si,Ge,Sn), Cr2Cu(Al,Ga,In),Mn2Fe(Si,GeSn), Mn2Co(Al,Ga,As). Many of the 26-electronsystems show gaps after 9 states, some show gaps after 12states, and a few show gaps after 14 states. None of thesesystems can generate the 8μB moment needed to generate ahalf-metal using the gap after 9 states. Several, however, cangenerate the moment of 2μB needed to generate a half-metalor near-half-metal by using a minority gap after 12 states. Amoment of 2μB would also be consistent with a half-metaltaking advantage of a majority gap after 14 states. In principle,a majority gap after 14 states and a minority gap after 12combined with a moment of 2μB could lead to a magneticsemiconductor. Mn2CoAl comes close to this situation.

V2Cu(P,As,Sb) have gaps in both spin channels after 9 statesand are nonmagnetic. V2CuSb also has pseudogaps in both spinchannels after 14 states.

V2Zn(Si,Ge,Sn) have only pseudogaps near 9 states and arealso nonmagnetic. None of the 26-electron systems with X =V are predicted to be half-metals or near-half-metals.

Cr2Co(P,As,Sb) have majority gaps after 9 states andminority gaps after 12 states. Cr2CoP also has a small minoritygap after 9 states that becomes a pseudogap for Cr2CoAs andCr2CoSb. Cr2CoP is a half-metal with a moment of 2μB .Cr2CoAs and Cr2CoSb are near-half-metals with momentsof 2.0047μB and 2.0016μB , respectively, however they areomitted from Table III because of their positive formationenergy.

Cr2Ni(Si,Ge,Sn) are remarkable for the large gaps after 9states in the majority channel. Cr2NiSi and Cr2NiGe also havesmaller gaps after 9 states in the minority channel. Cr2NiSiand Cr2NiGe also have gaps after 12 states in minority. TheFermi energy falls just above these gaps after 12 states givingmoments of 1.89μB and 1.94μB , respectively. Cr2NiGe alsohas a minority gap after 14 states. Cr2NiSn has a single gapafter 9 states in the majority channel and a relatively smallmoment (0.83μB). Cr2NiSi and Cr2NiGe satisfy our criteriafor near-half-metals and are listed in Table III.

Cr2Cu(Al,Ga,In) all show gaps after 9 states in both spinchannels. In addition, Cr2CuAl has gaps after 14 states inboth channels while Cr2CuGa has gaps after 14 states in theminority channel. For all cases, however, the net moments aretoo small to place the Fermi energy near any of the gaps. Thesethree systems are ferrimagnets with relatively large antialignedmoments on the two Cr atoms.

Mn2Fe(Si,Ge,Sn) all have minority gaps after 12 states.Mn2FeSi and Mn2FeSn also have pseudogaps after 9 states inmajority while Mn2FeGe has a gap after 9 states in minority.The moments (2.008μB , 2.016μB , and 1.999μB ) are theright size to make all three of these systems near-half-metals(Table III); however, the formation of Mn2FeSn is positive.

Mn2Co(Al,Ga,In) tend to have minority gaps (Al,Ga) orpseudogaps (In) after 12 states and their net moments tendto be large enough to take advantage of these gaps to makenear-half-metals. Thus, Mn2CoAl is a half-metal (Table II)and Mn2CoGa is a near-half-metal (Table III).

In addition to the gap in the minority channel after 12 statesthat Mn2CoAl utilizes to make itself a half-metal, it also has aslightly negative gap in the majority channel, i.e., a very slightoverlap between bands 14 and 15. The maximum of band 14 atthe � point is only slightly higher than the minimum of band 15at the X point. This situation has led some to refer this systemas a “spin-gapless semiconductor” [34,86]. In our calculationand also that of Ouardi et al. [28], it is a semimetal rather thana semiconductor.

27-electron systems. The 21 systems in our databasecomprise V2Zn(P,As,Sb), Cr2Ni(P,As,Sb), Cr2Cu(Si,Ge,Sn),Cr2Zn(Al,Ga,In), Mn2Fe(P,As,Sb), Mn2Co(Si,Ge,Sn),Mn2Ni(Al,Ga,As). A system with 27 electrons could takeadvantage of a minority gap after 12 states to make a half-metalwith a moment of 3μB or a majority gap after 14 states to tomake a half-metal with a moment of 1μB .

V2Zn(P,As,Sb): All three have gaps after 9 states in bothspin channels. V2ZnP is nonmagnetic while V2ZnAs andV2ZnSb have relatively small moments on the V sites withferrimagnetic alignment. Because their moments are small andthere are no gaps after 12 or 14 states, these systems are veryfar from being half-metals.

Cr2Ni(P,As,Sb): These three systems all generate large gapsafter 9 states in the majority channel, and gaps after 12 states

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and after 14 states in the minority channel. The gaps after 14states cannot be used to generate a half-metal because they arein the minority channel. The gaps after 12 states could be usedto make half-metals or near-half-metals but the net momentsare too small (2.03μB , 1.69μB , and 1.29μB , respectively) forthe Fermi energy to be near the gap.

Cr2Cu(Si,Ge,Sn): These three systems generate gaps after9 states in both channels and gaps (Si and Ge) or pseudogaps(Sn) after 14 states in the minority channel. There are no gapsafter 12 states. The moments are too small to take advantageof the gaps after 9 states and the gaps after 14 states are in thewrong channel to be useful for making half-metals.

Cr2Zn(Al,Ga,In): These systems only generate gaps after14 states; Cr2ZnAl in both channels, Cr2ZnGa in the minority,and for Cr2ZnIn there are no gaps. The one opportunity formaking a half-metal is the majority channel of Cr2ZnAl,but the net moment is somewhat smaller (0.706μB ) thanthe 1.0μB needed to make a half-metal. Pseudogaps near 9states can be identified in both majority and minority of allthree.

Mn2Fe(P,As,Sb): These systems all generate gaps after 12states in the minority channel and the moments have the correctvalue (3μB) to generate half-metals (Table II). The predictedhalf-metal Mn2FeSb is omitted from Table II because it ispredicted to have positive formation energy. These systemsare also predicted to have pseudogaps around 9 states in bothmajority and minority. Mn2FeP has a small gap after 9 statesin minority.

Mn2Co(Si,Ge,Sn): These systems all generate gaps after12 states in the minority channel. For Mn2CoSi and Mn2CoGethe moment is large enough (3μB) to generate a half-metal(Table II). The moment of Mn2CoSn is too small to generate agap. There are also gaps after 9 states in majority for Mn2CoSiand Mn2CoGe. This gap becomes a pseudogap for Mn2CoSn.

Mn2Ni(Al,Ga,As): No gaps are predicted for these threesystems. Mn2NiAl and Mn2NiGa have pseudogaps around 14states.

28-electron systems. The 18 systems with 28 valenceelectrons in our data set comprise Cr2Cu(P,As,Sb),Cr2Zn(Si,Ge,Sn), Mn2Co(P,As,Sb), Mn2Ni(Si,Ge,Sb),Mn2Cu(Al,Ga,In), and Fe2Co(Al,Ga,In). In principle, thesesystems could take advantage of gaps after 14 electrons tomake zero moment half-metals or semiconductors, however,we found no examples of such an electronic structure ina system with negative formation energy. They can alsotake advantage of a minority gap after 12 states to makehalf-metals with moments of 4μB per formula unit. We foundthree examples of this type of electronic structure.

Two systems Cr2Cu(P,As,Sb): These systems all havesizable gaps in majority after 9 states and smaller gaps inminority after 9 states. Cr2CuP has a pseudogap after 14 statesin minority and Cr2CuSb has a minority pseudogap after 14states. The moments are small.

Cr2Zn(Si,Ge,Sn): These systems have pseudogaps near 9states per formula unit in both spin channels. Cr2ZnSi andCr2ZnGe have gaps after 14 electrons. Cr2ZnSi has gaps inboth channels so that it is predicted to be a ferrimagneticsemiconductor with zero net moment and two different gapwidths. The smaller gap is a direct gap at the L point and ison the order of 0.02 eV so that XA phase Cr2ZnSi could be

called a “spin gapless semiconductor.” Cr2ZnGe only has a gapafter 14 states in one spin channel. Its net moment is almostzero (0.0004μB ) so that it is predicted to be a near-half-metal.Unfortunately, both of these interesting systems are predictedto have positive formation energy and hull distances in excessof 0.3 eV/atom.

Mn2Co(P,As,Sb): Mn2CoP and Mn2CoAs have gaps in bothchannels after 9 states. All three systems have gaps after 12states in minority. All three have the requisite moment (4μB)to generate half-metals and are listed in Table II. We includedetails about the calculated Mn2CoSb XA phase in Table IIeven though its calculated formation energy is positive becauseof the controversy over the experimentally observed phase asdiscussed in Sec. III A.

Mn2Ni(Si,Ge,Sb): These systems all have minority gapsafter 12 states. Mn2NiSi and Mn2NiGe also have majoritypseudogaps near 9 states. The moments on these systems areall too small to take advantage of the minority gap after 12states.

Mn2Cu(Al,Ga,In): One can identify pseudogaps near 9states in majority in Mn2CuAl and Mn2CuGa along with atiny gap after 9 states in minority in Mn2CuGa. One canalso identify pseudogaps near 14 states in both majority andminority in Mn2CuAl.

Fe2Co(Al,Ga,In): These systems have minority gaps after 9states. Pseudogaps near 12 minority states can also be identifiedabove the Fermi energy in all three systems. Even if there werea minority gap after 12 states, the moments in these systemswould be too large to take advantage of it.

29-electron systems. Our database contained 18 systemswith 29 valence electrons per formula unit: Cr2Zn(P,As,Sb),Mn2Ni(P,As,Sb), Mn2Cu(Si,Ge,Sn), Mn2Zn(Al,Ga,In),Fe2Co(Si,Ge,Sn), and Fe2Ni(Al,Ga,In). In principle, a systemwith 29 electrons per formula unit could utilize a minoritygap after 9, 12, or 14 states together with moments of 9μB ,5μB , or 1μB , respectively, to make a half-metal. In practice,we only find one 29 electron half-metal, Mn2CuSi, with amoment of 1μB .

Cr2Zn(P,As,Sb): These systems all showed gaps after 9states and 15 states in the minority channel. Cr2ZnP also hasa gap after 9 states in the majority channel and a pseudogapafter 14 states in minority. Cr2ZnAs also has a pseudogap near9 states and a gap after 14 states in majority. In principle, thegap after 14 states in the majority could be utilized to makea half-metal, but the moment is too small (0.47μB). Cr2ZnSbhas in addition to the minority gaps after 9 and 15 states, apseudogap near 9 states in majority. The gap after 15 statesobserved in the minority channel of Cr2Zn(P,As,Sb) is unusualand can be traced to a singly degenerate state at � that isusually band 15 or band 17 dropping below a triply degeneratestate.

Mn2Ni(P,As,Sb): These three systems have gaps (Mn2NiP)or pseudogaps [Mn2Ni(As,Sb)] near 12 states in the minoritychannel, but the moments are all much less than the 5μB

that would be necessary to make a half-metal. In addition,pseudogaps near 9 states in majority and 14 states in minoritycan be identified for Mn2NiP.

Mn2Cu(Si,Ge,Sn): All have gaps after 9 states in both themajority and minority channels. Mn2CuSi also has a minoritypseudogap near 12 states and a minority gap after 14 states.

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The system uses the minority gap after 14 states to make ahalf-metal (Table II).

Mn2Zn(Al,Ga,In): All have small moments and no gaps.Pseudogaps in both channels near 9 states can be identified.

Fe2Co(Si,Ge,Sn): All have moments near 5μB . However,instead of gaps after 12 states in minority there are pseudogaps.All three have gaps after 9 states in minority. Fe2CoSi also hasa pseudogap near 9 states in majority.

Fe2Ni(Al,Ga,In): Also, all have moments near 5μB , but theonly gaps are after 9 states in the minority channel. Fe2NiAl hasa minority pseudogap near 12 states and majority pseudogapsnear 14 and 16 states. Fe2NiGa has majority pseudogaps near9 states and 16 states.

30-electron systems. The 15 systems in our databasewith 30 electrons are Mn2Cu(P,As,Sb), Mn2Zn(Si,Ge,Sn),Fe2Co(P,As,Sb), Fe2Ni(Si,Ge,Sn), and Fe2Cu(Al,Ga,In). Inprinciple, gaps after 14, 12, or 9 states could be used to makehalf-metals with moments of 2μB , 6μB , or 12μB , respectively.In practice, we find one negative-formation-energy near-half-metal with a moment near 2μB .

Mn2Cu(P,As,Sb): These all have gaps after 9 states in bothmajority and minority. Mn2CuP also has a gap after 14 states inthe minority channel. The moment is nearly 2μB which makesit a near-half-metal.

Mn2Zn(Si,Ge,Sn): These systems have pseudogaps in bothspin channels near 9 states. Mn2ZnSi also has a gap after 14electrons in the minority channel and a spin moment of exactly2μB which makes it a half-metal. Unfortunately, its calculatedformation energy is positive. Mn2ZnGe and Mn2ZnSn do nothave gaps in the DOS at the experimental lattice constant. If,however, their lattices are significantly contracted, the gap after14 states reappears and the Fermi energy falls into the gap.

Fe2Co(P,As,Sb): These all have gaps after 9 electrons inthe minority channel and pseudogaps near 12 states also inthe minority channel. The moments are large enough (5.76μB ,5.95μB , 5.93μB , respectively) to place the Fermi energy inthe vicinity of the 12-state pseudogap, but because there is notreally a gap, these are not near-half-metals.

Fe2Ni(Si,Ge,Sn): These all have gaps after 9 states in theminority. Fe2NiSi and Fe2NiGe also have majority gaps after9 states. Fe2NiSi also has a minority gap after 12 stateswhile Fe2NiGe has a minority pseudogap after 12 states.The moments are relatively large (4.78μB , 5.09μB , 5.23μB ,respectively) but not large enough (6μB would have beenrequired) to make them half-metals or near-half-metals.

Fe2Cu(Al,Ga,In): These all have relatively large minoritygaps after 9 states and small majority gaps after 14 states(Fe2CuAl and Fe2CuGa) or after 16 states (Fe2CuIn). Fe2CuInalso has a pseudogap after 14 states in the majority. Themajority gaps after 14 states cannot be used to make a half-metal and although the majority gap after 16 could be used,the moment is too large, 4.84μB rather than 2μB .

31-electron systems. The 15 systems in our databasewith 31 electrons comprise Mn2Zn(P,As,Sb), Fe2Ni(P,As,Sb),Fe2Cu(Si,Ge,Sn), Fe2Zn(Al,Ga,In), and Co2Ni(Al,Ga,In).Compounds with 31 electrons could, in principle, take ad-vantage of a minority gap after 14, 12, or 9 states with amoment of 3μB , 7μB , or 13μB , respectively. In addition,we find one example of a gap after 15 states which wouldallow a half-metal with a moment of 1μB . In practice, we find

no 31-electron half-metals or near-half-metals with negativeformation energy.

Mn2Zn(P,As,Sb): All three systems have gaps after 9 statesin minority. Mn2ZnP also has a gap after 9 states in majoritywhile Mn2ZnAs has a pseudogap near 9 states in majority.Mn2ZnP has a pseudogap near 14 states in minority. Mn2ZnAshas a tiny minority gap after 15 states and a moment (1.001μB )that places the Fermi energy very near this gap. Unfortunately,its formation energy is positive. Mn2ZnSb has a similarelectronic structure and moment, but the tiny gap in Mn2ZnAsis replaced by a pseudogap in Mn2ZnSb.

Fe2Ni(P,As,Sb): These all have minority gaps after 9 states.Fe2NiP also has a minority gap after 12 states while Fe2NiAsand Fe2NiSb have pseudogaps after 12 states. The momentsare too small to take advantage of the gap or pseudogaps after12 states.

Fe2Cu(Si,Ge,Sn): These all have gaps after 9 states in bothspin channels, with the minority gap being larger than themajority gap. In addition, Fe2CuSi and Fe2CuSn have narrowgaps after 14 states in the majority. None of these gaps are nearthe Fermi energy.

Fe2Zn(Al,Ga,In): These systems have gaps (Al) or pseu-dogaps (Ga,In) at 14 states in minority. The moments are toolarge to place the Fermi energy near these gaps or pseudogaps.

Co2Ni(Al,Ga,In): These systems have numerous gaps andpseudogaps. All three have narrow, deep, pseudogaps near 14states in minority and have moments (2.99, 3.02, 3.13)μB thatplace the Fermi energy in the pseudogaps. They are not “near-half-metals” according to our definition. In addition, all threehave minority pseudogaps near 12 states. They all also havegaps after 9 states in minority. Co2NiAl has a gap after 14 statesin majority. Co2NiGa has a gap after 9 states in majority andpseudogaps near 14 and 16 states in majority.

32-electron systems. The 12 systems with 32 electronsin our database include Fe2Cu(P,As,Sb), Fe2Zn(Si,Ge,Sn),Co2Ni(Si,Ge,Sn), and Co2Cu(Al,Ga,In).

Fe2Cu(P,As,Sb): These systems all have gaps after 9 statesin both spin channels and pseudogaps after 14 states in theminority channel. Their moments are all greater than 4μB , andthe moment of Fe2CuP is 4.018μB , quite close to the valueof 4 needed to generate a half-metal. Unfortunately, the Fermienergy is on the edge of a pseudogap rather than a gap.

Fe2Zn(Si,Ge,Sn): The 31-electron Fe2ZnZ XA compoundshave pseudogaps rather than gaps near 9 states in both spinchannels, but similar to the 31-electron Fe2CuZ systems, theyhave pseudogaps near 14 states in the minority channel. Thepseudogap near 14 states in Fe2ZnSi is remarkable in that theDOS plots appear to show a gap after 14 states. Its moment(3.982μB ) is close enough to 4 that we would classify Fe2ZnSias a near-half-metal and include it in Table III because it alsohas a negative formation energy. However, bands 14 and 15cross between X and W , spoiling the gap.

Co2Ni(Si,Ge,Sn): These all have gaps after nine states inboth spin channels and pseudogaps near 12 states in minority.Their moments are too small (2.51μB , 2.56μB , 2.70μB ) toplace the Fermi energy near the pseudogaps. None are half-metals or near-half-metals.

Co2Cu(Al,Ga,In): These all have gaps after 9 states inthe minority channel. None are half-metals or near-half-metals.

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33-electron systems. The 12 systems with 33 electronsin our database include Fe2Zn(P,As,Sb), Co2Ni(P,As,Sb),Co2Cu(Si,Ge,Sn), and Co2Zn(Al,Ga,In). These systems wouldneed a moment of 5μB to utilize a gap after 14 statesto make a half-metal. They have neither the gaps nor themoment.

Fe2Zn(P,As,Sb): These systems have either gaps (P,As) orpseudogaps after 9 states in both spin channels.

Co2Ni(P,As,Sb): These systems all have gaps after 9 statesin both spin channels. Pseudogaps near 12 states in minoritycan also be identified for all three systems.

Co2Cu(Si,Ge,Sn): These systems also have gaps after 9states in both channels. Two (Si,Ge) have pseudogaps after12 states in both spin channels.

Co2Zn(Al,Ga,In): No gaps or convincing pseudogaps wereobserved in these systems.

34-electron systems. The 9 systems in our database with34 electrons include Co2Cu(P,As,Sb), Co2Zn(Si,Ge,Sn), andNi2Cu(Al,Ga,In).

Co2Cu(P,As,Sb): These systems have gaps after 9 states inboth channels. Co2CuP also has a majority gap after 12 stateswhich becomes a pseudogap for the other two compounds. Themoments are not nearly large enough to place the Fermi energynear the gaps or pseudogaps.

Co2Zn(Si,Ge,Sn): These systems all have pseudogaps near9 states in both spin channels and also a pseudogap near 12states in the minority channel.

Ni2Cu(Al,Ga,In): These systems are nonmagnetic and haveno gaps. Ni2CuAl and Ni2CuGa have pseudogaps near 12 statesin both channels. Ni2CuGa and Ni2CuIn have pseudogaps after9 states in both channels.

35-electron systems. The 9 systems in our database with35 electrons include Co2Zn(P,As,Sb), Ni2Cu(Si,Ge,Sn), andNi2Zn(Al,Ga,In). A system with 35 electrons would need amoment of 7μB to take advantage of a gap after 14 states inminority. Such moments would be unlikely in these systemsbecause the d bands are nearly filled for Co and especially forNi.

Co2Zn(P,As,Sb): These systems are magnetic with mo-ments relatively small moments (2.4μB , 2.5μB , and 2.7μB).Co2ZnP and Co2ZnAs have gaps after 9 states in both spinchannels while Co2ZnSb has pseudogaps near 9 states in bothspin channels. Pseudogaps near 14 states can be identified inall three systems.

Ni2Cu(Si,Ge,Sn): These are all nonmagnetic and have gapsafter 9 states in each of the identical spin channels. Pseudogapsnear 12 states can be identified in all three as well as near 14states in Ni2CuSn.

36-electron systems. The 6 systems in our database with 36electrons include Ni2Cu(P,As,Sb) and Ni2Zn(Si,Ge,Sn). Noneof these systems are magnetic.

Ni2Cu(P,As,Sb): These have gaps after 9 states and pseu-dogaps near 12 states and 14 states.

Ni2Zn(Si,Ge,Sn): These have pseudogaps near 14states.

37-electron systems. The 6 systems in our database with 37electrons are Ni2Zn(P,As,Sb) and Cu2Zn(Al,Ga,In). None ofthese 6 systems have a magnetic moment and therefore cannotbe half-metals.

Ni2Zn(P,As,Sb): Ni2ZnP and Ni2ZnAs have gaps after 9states. Ni2ZnSb has a pseudogap near 9 states. Pseudogapsafter 14 states can be identified for all three.

Cu2Zn(Al,Ga,In): These systems have no gaps.38-electron systems. The three systems in our database with

38 electrons are Cu2Zn(Si,Ge,Sn). They have neither momentsnor gaps.

39-electron systems. The three systems in our database with39 electrons are Cu2Zn(P,As,Sb). They have neither momentsnor gaps.

V. CONCLUSION

We have performed extensive first-principles calculationson 405 inverse Heusler compounds X2YZ, where X = Sc, Ti,V, Cr, Mn, Fe, Co, Ni, or Cu; Y = Ti, V, Cr, Mn, Fe, Co, Ni, Cu,or Zn; and Z = Al, Ga, In, Si, Ge, Sn, P, As, or Sb with a goal ofidentifying materials of interest for spintronic applications. Weidentified 14 semiconductors with negative formation energy.These are listed in Table I. We identified many half-metals andnear-half-metals. Those half-metals and near-half-metals thatwe identified and that have negative formation energy are listedin Tables II and III, respectively.

We have identified four half-metals and six near-half-metalsthat meet a criterion for stability that we estimated based ona study of the hull distances of known XA phases. Seven ofthese ten predicted phases have not (to our knowledge) beenobserved experimentally.

In establishing our stability criterion, we analyzed the ex-perimental and theoretical data for several reported XA phases(Mn2CoIn, Mn2CoSb, Mn2CoSn, Mn2RuSn) and tentativelyconcluded that they are not XA but are more likely a disorderedphase such as L21B. We recommend more extensive studiesof these materials.

The 10 half-metallic or near-half-metallic phases, that weidentified as reasonably likely to be stable, are all Slater-Pauling phases, i.e., they have a gap in the density of statesafter three states per atom in one of the two spin channels,and the Fermi energy is in or near this gap. Many half-metallicand near-half-metallic phases based on a gap after 9 states performula unit (2.25 states per atom) were identified, but weestimate that other phases or combination of phases are likelyto be more stable in equilibrium.

ACKNOWLEDGMENTS

The authors acknowledge financial support from the Na-tional Science Foundation through Grants No. DMREF-1235230 and No. DMREF-1235396. J.H. and C.W. (OQMDstability and SQS calculations) acknowledge financial supportvia ONR STTR Grant No. N00014-13-P-1056. The authorsalso acknowledge Advanced Research Computing Services atthe University of Virginia and High Performance Computingstaff from the Center for Materials for Information Technol-ogy at the University of Alabama for providing technicalsupport that has contributed to the results in this paper. Thecomputational work was done using the High PerformanceComputing Cluster at the Center for Materials for Information

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Technology, University of Alabama, resources of the NationalEnergy Research Scientific Computing Center (NERSC), aDOE Office of Science User Facility supported by the Office

of Science of the U. S. Department of Energy under ContractNo. DE-AC02-05CH11231 and the Rivanna high-performancecluster at the University of Virginia.

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